Week-14

Decision Sciences Volume 47 Number 3 June 2016

© 2015 Decision Sciences Institute

Decision Making in Cross-Functional Teams: The Role of Decision Power*

Zhijian Cui† IE Business School, Calle de Maria de Molina, 12, Madrid 28006, Spain, e-mail: Zhijian.cui@ie.edu

ABSTRACT

Through a series of game-theoretical models, this study systematically examines de- cision making in cross-functional teams. It provides a framework for the design of an organization-specific decision-making process and for the alignment of a team’s mi- crodecision with the “optimal” decision that maximizes the firm’s payoff. This study finds that even without changing the team leader, firms could change and even dictate the team’s microdecision outcome via adjusting the team member’s seniority, empowering team members with veto power or involving a supervisor as a threat to overrule the team decision. This finding implies that to reposition products in the marketplace, structur- ing cross-functional teams’ microdecision-making processes is essential. [Submitted: January 13, 2014. Revised: September 27, 2014. Accepted: December 6, 2014.]

Subject Areas: cross-functional team, decision-making process, decision power, game theory, and stakeholder management.

INTRODUCTION

Many critical business decisions are currently made by cross-functional teams composed of representatives from multiple functionalities or departments (Griffin, 1997; Hutchison-Krupat & Kavadias, 2013; Mihm, 2010). In the context of new product development (NPD), for example, firms rely on cross-functional teams for product development between 70% and 75% of the time (McDonough, 2000). The widespread use of cross-functional teams is a result of the overwhelming success attributed to them: cross-functional teams improve product quality by in- corporating market information earlier in the design process (Terwiesch, Loch, & De Meyer, 2002), reduce production costs by rendering products more “manufac- turable” (Lovelace, Shaprio, & Weingart, 2001) and decrease development time and rework (Griffin, 1997). Conversely, numerous studies also discuss the “dark side” of cross-functional teams. Because of their diverse backgrounds, expertise, access to information and so forth, cross-functional team members often have

*The author gratefully acknowledges the Associate Editor and two anonymous referees for their highly constructive feedback during the review process. The author is also grateful for the grant from Spanish Ministry of Economy and Competitiveness (ECO2013-48403-R) to support this research project.

†Corresponding author.

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different opinions about “what should be done” (Atuahene-Gima & Evangelist, 2000; Griffin & Hauser, 1996; Pinto, Pinto, & Prescott, 1993; Mihm, 2010). Con- sequently, team members’ objectives are often “misaligned” with firms’ overall objectives and priorities (Hutchison-Krupat & Kavadias, 2013; Mihm, 2010). Such misalignment, combined with the specific micro decision-making structure, may lead to “suboptimal” decision outcomes.

One case that motivates our study involves the development process of BMW’s 7-series sedan (Pisano, 2002). In developing the 7-series sedan, BMW strategically planned to narrow the gap between its product and those of its Japanese competitors by improving its product conformance quality and reducing its manu- facturing costs. BMW possessed both a clear technology strategy and well-trained design engineers. Unfortunately, because of strong organizational resistance, im- plementing BMW’s repositioning objective was not a trivial task. At the level of micro decision making, design engineers played a dominant project-leadership role in BMW’s traditional design system, and their maximal flexibility was ensured by the existence of a specialized prototype shop, general-purpose tooling and the power to revise designs until several days before manufacturing began. In contrast, manufacturing engineers were nearly excluded from the design process, and their concerns regarding the ease and cost of manufacturing were rarely incorporated into the product design. Even worse, the manufacturing engineers were not able to learn about the detailed assembly process. Thus, they had to set up production “from scratch,” and the 7-series sedan’s conformance quality, measured as the likelihood of defect, was inevitably sacrificed.

Another example that motivates this study comes from a field study we con- ducted at Nokia, a company famous for its cutting-edge hardware technologies in the cell phone industry. In the past, Nokia allocated the majority of its financial resources to hardware development. As the smart-phone market grew, its market share began to drop, and its top management realized the strategic importance of changing the focus of its R&D expenditures from developing hardware to develop- ing software, such as applications and services. However, mid-level managers were skeptical of this strategic initiative. In conversations with the authors, one senior executive from the marketing functionality department quipped, “Our company has historically been dominated by stubborn engineers, who do not care and do not even know what today’s customers really need.” Ironically, one hardware design engineer complained, “We cannot let important decisions be made by people with ‘soft’ minds.... Those people only care about customers; they do not always under- stand technologies.” After several years of struggle, Nokia finally lost its dominant position in the cell phone market.

The above examples show that a company in which design engineers enjoy decision power will likely promote a design focus, with the potential downside of excessive cost, whereas a manufacturing-focused organization may achieve mini- mum cost at the expense of an inferior design output. Thus, the “optimal” decision that a firm intends to pursue may be “misaligned” with the actual decision out- come from a cross-functional team, and a simple “over-the-wall” approach (Adler, 1995) is insufficient to ensure that “correct” decisions are made within a complex organization. In addition, both examples show that such “misalignment” does not result from insufficient resources, such as expertise and machinery. How does misalignment arise, then? Is it simply a result of bad management or an indicator

494 Decision Making in Cross-Functional Teams

of improper decision-making processes or even improper organizational design? Moreover, are decision-making processes universally “proper” for all companies, or do they depend on a company’s specific organizational characteristics? From the perspective of a firm’s top management, what levers can companies use to minimize such misalignment and thus mitigate the likelihood of “suboptimal” outcomes?

Motivated by these challenges and questions, this paper aims to build a game- theoretical framework that links team members’ preferences and decision power that stem from both personal attributes and participation at certain stages of the mi- cro decision-making process. Our results show when and why management should use different levers (aligning preferences, adjusting team composition, empower- ing team members with different types of decision power, and involving senior management) to align a cross-functional team’s micro decision with the “optimal” decision that maximizes the firm’s payoff. Our study therefore establishes a de- tailed map between cross-functional teams’ micro decision-making processes and decision outcomes and demonstrates the importance of managing and designing the detailed structure of decision making.

In the following, we first review the literature and position our study within the existing research. Then we describe the model setups and examine the impact of team members’ preferences on a team’s decision outcomes and strategic misalign- ment. Considering team members’ decision power that stems from both personal attributes and participation in the decision-making process, we further examine the team’s decision outcome under different settings in which the decision-making structure is slightly altered. We conclude the study and discuss its implications in the last section.

LITERATURE REVIEW

Prior research has established that conflicts among different firms or divisions within a firm are caused by their fundamentally different interests (Balasubra- manian & Bhardwaj, 2004; Hutchison-Krupat & Kavadias, 2013; Li & Atkins, 2002; Mihm, 2010; Narayanan & Raman, 2004; Siemsen, 2008). For example, a firm’s manufacturing division incurs manufacturing costs and therefore primarily aims to minimize costs (Li & Atkins, 2002), whereas a firm’s marketing division aims to maximize revenue (Natter, Mild, Feurstein, Dorffner, & Taudes, 2001; Balasubramanian & Bhardwaj, 2004). However, the marketing division’s pricing policy, which maximizes its expected revenue, also increases the firm’s expected leftover inventory, the cost of which is borne by the manufacturing division (Li & Atkins, 2002). Thus, one party’s interest is maximized at the expense of another party’s interest through a product-pricing policy that incurs excessive inventory costs (Li & Atkins, 2002). In the references that examine inter-firm or -division coordination, each party has its own interest and can make individual decisions to maximize its interest. In the context of cross-functional teams, however, team members share a fundamentally common interest, and team members’ differences in objectives or predictions of decision outcomes are caused by their differences in either the available information or attributions of causality (Groves, 1973). In addition, all decisions in teams are jointly made by team members under a par- ticular decision-making structure (Griffin, 1997; Griffin & Hauser, 1996). These

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team-specific settings (i.e., in which team members have a common interest but different objectives) constitute the point of departure for our study and require a modeling approach that differs from that of prior research.

Existing studies on cross-functional teams have empirically studied the ef- ficacy of management “levers” in mitigating conflicts among project members. These management levers include aligning team members’ preferences via job rotations and team-building training sessions (Griffin & Hauser, 1996), facilitating communication (Griffin & Hauser, 1996; Henke, Krachenberg, & Lyons, 1993), and building trust (Henke et al., 1993; Natter et al., 2001) and ensuring team auton- omy (Griffin & Hauser, 1996). However, these levers are often studied in isolation, and it remains unclear how these levers may interact with each other and the condi- tions under which they can be jointly applied. Our study makes two contributions to this stream of literature. First, we introduce decision power, which has been rarely examined in prior research, as a management lever to align a team’s micro decision with the firm’s objective. Second, we build a framework that considers several management levers and provide new insights into the efficacy of some well-established management levers from the perspective of strategic alignment.

Existing studies in the NPD literature have analytically examined the de- sign of mechanisms to align a team’s micro decisions with the firm’s objec- tives. One mechanism that is often mentioned is monetary incentives, such as transfer payments within an organization (Mihm, 2010) or a codevelopment con- tract between collaborative organizations (Bhaskaran & Krishnan, 2009; Savva & Scholtes, 2014). In a general setting, Mihm (2010) shows that by the revelation principle, an optimal Bayesian contract can be constructed between a principal (firm) and multiple agents (engineers) such that all engineer-agents with private information will behave more in line with the firm’s interests. Another mechanism is performance measurement (Hutchison-Krupat & Kavadias, 2013; Mihm, 2010; Siemsen, 2008). This stream of literature primarily aims to tackle the challenge of managing agents’ nonverifiable efforts in complex organization by awarding (pe- nalizing) individual employee actions that are aligned (misaligned) with a firm’s overall task priority. For example, Mihm (2010) shows the value of target cost- ing in aligning design engineers’ individual objectives with the firm’s objectives. Hutchison-Krupat and Kavadias (2013) studies the conditions in which offering performance-based contracts is beneficial to organizations. Our study takes a dif- ferent approach from that of previous studies set forth above. We abstract out the challenges of nonverifiable efforts in cross-functional teams and instead focus on examining the impact of the team’s decision-making structure on the team’s decision outcome and the magnitude of misalignment.i

Recently, a stream of NPD literature has emerged to account for the hierarchi- cal nature of organizational decision making (Chao & Kavadias, 2008; Hutchison- Krupat & Kavadias, 2013; Mihm, Loch, Wilkinson, & Huberman, 2010). These studies primarily consider the vertical distribution of decision power between the principal (firm) and its agent (the division). For example, Hutchison-Krupat and

i In the following section, we provide more discussions on how monetary incentives could be implemented for alignment.

496 Decision Making in Cross-Functional Teams

Kavadias (2013) compares the efficacy of a decision-making process that dictates resource levels (top down) with that of one that delegates resource decisions and relinquishes control (bottom up). In their setting, the tradeoff that a principal (firm) must make is whether to delegate decision power to the division in exchange for more precise information or to retain decision power to achieve more efficient resource planning. Our study differs from Hutchison-Krupat and Kavadias (2014) in two important aspects. First, motivated by the BMW challenge (Pisano, 2002), the firm is assumed to have a better vision of “what should be done” than the team members in our study; by contrast, in Hutchison-Krupat and Kavadias (2013), the agent (division) has more precise information than the principal (firm). Therefore, our study aims to reduce the misalignment between a cross-functional team’s mi- cro decision and the firm’s macro objectives via different organizational levers, whereas Hutchison-Krupat and Kavadias (2013) aims to show when it is optimal to implement a “top-down” vs. “bottom-up” strategy. Second, in our models, de- cision power is horizontally distributed among team members in a team, whereas in Hutchison-Krupat and Kavadias (2013), decision power is vertically distributed from the headquarters to the divisional level.

Our study is also related to, yet different from, the economics literature on decision rights (Aghion & Tirole, 1997; Grossman & Hart, 1986; Hart & Moore, 1990). In our study, the members of cross-functional teams do not have the right of ownership (e.g., Grossman and Hart, 1986; Hart and Moore, 1990). Instead, team members can obtain decision power through personal attributes or participation in specific stages of the decision-making process. Therefore, our study implicitly models team members’ real authority in the organization, whereas in Aghion and Tirole (1997), an agent’s incentive to invest in information gathering is only affected by his or her formal authority.

Finally, our study contributes to the literature on strategy implementation. Similarly to ours, studies in this stream of literature suggest that a particular type of “alignment” or “congruence” within an organization is essential to a firm’s success (we refer readers to Yang, Sun, and Eppler (2010) for a comprehensive review of this literature stream). To implement a strategy, firms must also use several tactics, such as persuasion, teamwork, negotiation, goal commonality, and total quality management (Nutt, 1989). However, these tactics are inevitably ad hoc, and their applicability heavily depends on managers’ personal characteristics and management skills (Nutt, 1989; Sashittal & Wilemon, 1996). By contrast, our study presents a game-theoretical framework to show how a change in a team’s decision-making structure can systematically alter the team’s decision outcome.

MODEL SETUPS AND SOURCES OF DECISION POWER

In this section, we formulate a simple and intuitive game-theoretical model to capture the decision-making process in a cross-functional team. First, we describe the decision value as a single dimensional space f, where f � 0. In the context of NPD, the decision to be made, for example, the product feature parameter, can be a multiple-dimensional space (Srinivasan, Lovejoy, & Beach, 1997), as a product usually contains multiple features, such as weight, size, shape, manufacturing cost, the number of functionalities, etc. However, these features are rarely independent

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of each other. For example, increasing product size creates extra space in which to install additional functionalities, which, in turn, increases the product’s weight and manufacturing cost. Thus, given a fixed R&D budget, the decision-making in NPD also involves tradeoffs between different performance requirements (Ulrich & Eppinger, 2003). We can therefore collapse a multi-dimensional decision space into a critical, one-dimensional decision value, and the decision values of other dimensions can be simply mapped as a function of this critical decision. This method is commonly applied within the NPD literature (e.g., Terwiesch & Loch, 2004; Lacourbe, Loch, & Kavadias, 2009; Williams, Kannan, & Azarm, 2006).

Second, the cross-functional team in our study is composed of two parties, and each party has one specific team responsibility. For example, one party is in charge of collecting vital information about market needs, whereas another creates a detailed product design based on technology feasibility and customer needs. In contrast to the situation of firms involved in inter-firm collaborations, team members must share a fundamental common interest (because they belong to the same team), but they have different preferences because of their access to different information (Groves, 1973). Thus, the two parties have symmetric utility functions (reflecting a fundamental common interest) but different optimal solutions (i.e., the utility functions shift relative to each other).

To echo the Nokia example mentioned above, the tension between two parties is assumed to arise from the budget allocation for developing hardware. In partic- ular, one party (we refer to it as party A) believes that the quality of a product’s hardware performance is vital for market success and thus favors an R&D budget that is primarily allocated to hardware development. In contrast, another party (we refer to it as party B) believes that software performance plays a more important role than hardware performance in product success. Thus, party B favors a budget that allocates a greater proportion of funds to software development. To focus on examining the impact of decision power, our model abstracts the team members’ moral hazard effect. In other words, the team members honestly use the budget to develop the product only. Therefore, there is a one-to-one mapping between the budget and the realized product performance (i.e., a higher budget for software development leads to higher software performance, and vice versa). Therefore, the expected utility of each party can be transformed into a function of the single decision value f, and we let each party’s utility function be a quadratic function of f ii. Therefore, the payoffs of the two parties can be described as follows:∏

A

= P − a ( f − f ∗A

)2 . (1)

∏ B

= P − a ( f − f ∗B

)2 . (2)

ii In the literature of NPD and design engineering, the quadratic utility function has been widely used (Bhaskaran and Krishnan, 2009; Williams et al., 2006). In fact, we can show that the shifted quadratic utility representation is quite general. Specifically, it can capture both a vertical (which drives costs) and a horizontal product feature (which is cost-neutral, such as color). The formal analysis is available upon request.

498 Decision Making in Cross-Functional Teams

Figure 1: Firms’ and team members’ optimal decision values.

Here, P denotes the maximum payoff potential from making the decision, and f ∗

i denotes each party’s perceived “optimal” decision value that maximizes

the expected payoff. Similarly, the firm’s payoff function also takes a quadratic form, and it can be described as follows:∏

F

= P − a ( f − f ∗

)2 . (3)

where f� denotes the truly “optimal” decision value that maximizes the firm’s expected payoff.

The term a captures the sensitivity of the decision’s payoff with respect to the decision value. For simplicity, we let a =1 throughout this study. To exclude the trivial case, let f ∗ �= f ∗

A �= f ∗

B . Without a loss of generality, let f ∗

A > f ∗

B . The

utility functions of the firm and the two team members are plotted in Figure 1. One may argue that the sharp focus on the effect of each party’s utility

on software and hardware performance may seem limiting because both utilities usually depend on both hardware and software. Thus, alternatively, we consider the case in which both parties have weight in affecting software and hardware performance such that their utilities are jointly concave for both software and hardware performance. Note that this model formulation encompasses cases of convex combinations of party A and B’s utility. With a few additional minor technical assumptions, all of the results in this paper can be replicated. Thus, our results are robust to relatively drastic changes in model formulations.In an organization, stakeholders can gain decision power from two sources: (1) personal attributes, such as reputation, social status, and experience (Mintzberg, 1979; Pfeffer, 1992); and (2) structural sources, such as a stakeholder’s participation in the decision-making process (Atuahene-Gima & Evangelist, 2000; Pfeffer, 1992). To quantify the decision power that arises from their personal attributes, we borrow a method from bargaining theory (Muthoo, 2002) and define party A (B) to have an outside option or fallback utility uA (uB ). Higher ui implies stronger personal attributes and therefore more decision power (Pfeffer, 1992). In cross-functional teams, a team member’s outside option denotes his or her potential payoffs from other projects in the project portfolio (Chao & Kavadias, 2008) as he or she releases resources for the focal project.

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To account for decision power that stems from structural sources, we must examine in detail the role played by a team member in the decision-making process. In the context of supply chain management, the Stackelberg game is a well- established approach to model leader-follower competition between firms (Li & Atkins, 2002). In the context of cross-functional teams, however, the Stackelberg game is not a suitable model to represent a team leader’s decision power. First, in a Stackelberg game, the “leader’s” decision power solely results from his or her first-mover advantage, and the follower’s (i.e., second mover’s) action can only be the best response to the first mover’s action. In cross-functional teams, however, a team leader may not always move first, and his or her power may instead arise from the participation in a critical stage of decision making. Second, parties in a Stackelberg game make individual decisions to maximize their own payoffs, whereas in cross-functional teams, all members are bounded by one common team decision, which can be a joint decision agreed upon by all parties or a decision made and enforced by the team leader. Therefore, the unique nature of decision making in cross-functional teams requires a different modeling approach.

According to the seminal work by Mintzberg (1979), a decision-making process can be described as a sequence of stages that includes collecting infor- mation, processing that information to present advice, choosing what is to be implemented, and authorizing the decision, which may mean vetoing the proposed choice. We refer to the resulting types of decision power, which depend on the control over these various stages, as information, initiation or advice, veto, and choice power. Among the four types of decision power that arise from participa- tion in the decision-making process, choice power differentiates a leader from a follower (Mintzberg, 1979). In cross-functional teams, information and initiation power cannot be treated in isolation because initiation (advice) power is futile when it is accorded to a party who is not aware of critical information. A team member could also raise objections to or “veto” another member’s plans and thus “turn down” a proposal that counters his or her preference. In this study, we let party A (B) be the team leader (follower). The team leader (party A) owns choice power and may also own three other types of decision power depending on the participation of the team follower (party B).

In the following sections, we examine the impact of three organization- specific factors on a team’s decision outcome and the magnitude of misalignment with the firm’s interest: (1) the discrepancy between the preferences of team members, (2) decision power that arises from their personal attributes, and (3) decision power that arises from participation in the decision-making process.

THE ROLE OF TEAM MEMBERS’ PREFERENCES IN DECISION MAKING

To facilitate the analysis, we first define several notations. Appendix A summarizes all of this study’s assumptions and parameters. Define for party i of type t, where i � {A,B} and t denote the party’s optimal decision value f t∗

i , the set of decision

values �t i

that the party prefers to receive his or her outside option, i.e., �t i

defines a range in which potential mutual agreement is acceptable for party i, which we

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Figure 2: Compromise potential for party B of type t when preference varies.

term “party i’s compromise potential.” Figure 2 illustrates party B’s compromise potential �t

B , where party B’s preference can B only two values (t � {A,B}). The

limit f t B

is the right endpoint of �t B

which represents the “best” decision value for party A from party B’s compromise potential.

Using the simple formulations of our model, we can immediately categorize two extreme scenarios. First, when the preferences of team members are “close” enough (for example, team leader A’s optimal

decision value f ∗ A

lies within team follower B’s compromise potential �t B

), the team leader can simply assert his or her preferred design (f ∗

A ) and knows that

the team follower (party B) will always accept. In this case, the decision value will be set at f ∗

A , and the magnitude of misalignment is | f ∗

A −f ∗|. By contrast,

when the parties’ preferences are too divergent such that the team leader will receive less of a payoff from working on the project than from pursuing his or her outside option (P − (f t

B − f ∗

A )2 < uA), the team leader would rather abandon

the project or at least postpone making any team decisions even though such an outcome is “unwanted” from the firm’s perspective. In this case, firms could consider using traditional project management methods, such as organizing team- building training sessions or job rotations to align team members’ preferences (mathematically equivalent to minimizing | f t

j − f ∗

i |, where j �{A,B} and j �i ).

Between these two extreme scenarios, we define the third case, in which party i’s most favorable decision value f ∗

i is not acceptable to party j, and thus, bargaining

is indeed necessary (we call this condition the “no easy win-win” condition) in which a mutually acceptable solution could be found, given that party i has outside option ui (we call this condition the “common ground” condition). In this scenario, one general conclusion immediately follows:

Lemma 1: A set of decision values always exists such that both parties prefer any one of these decisions to receiving an outside option ui when two conditions hold:

Condition1 (no easy win − win) : f ∗i /∈ �tj Condition 2 (common ground) : �i

( f

t j

) ≥ ui

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Figure 3: Compromise potential for party B of type t when the outside option varies.

We provide detailed proofs in Appendix C. Formally, Lemma 1 states that under the above two conditions, there exists a subset of decision value f (f � (f t

′ A

, f t

B ), where t and t’ denote the type of party B and A, respectively) such that for

each outcome f in this set, �A(f ) > uA and �B (f ) > uB . Lemma 1 shows the existence of feasible decision outcomes from the cross-functional team and further defines the range of these outcomes even though it does not indicate which exact decision the team would make or the magnitude of strategic misalignment.

THE ROLE OF DECISION POWER IN DECISION MAKING

In the following analysis, we examine in detail how firms could use decision power that arises from both personal and structural sources as management levers to influence a team’s decision outcome and strategic misalignment.

The Role of Decision Power that Arises from Personal Attributes

Given the preferences of both the team leader (A) and the team follower (B), we now vary team follower B’s outside option ut

B . We continue to use the notations defined

before, except we redefine the set of decision values �t B

that B prefers to receive his or her outside option as �t

B = {f : �t

B (f ∗

B ) ≥ ut

B }. Figure 3 illustrates party B’s

compromise potential �t B

, where party B’s outside option can be only two values (t � {H, L}). Clearly, as B’s outside option increases, his or her compromise potential shrinks, and his or her stance related to the decision value becomes tougher (i.e., f H

B < f L

B ). Figure 3 shows in a straightforward fashion that as uB

becomes too low, the team leader can do whatever he or she desires. In contrast, as uB becomes too high (for example, by introducing a very senior team follower), P − (f t

B − f ∗

A )2 < uA, and thus, no team decision can be expected. In this case,

firms could consider adjusting the team composition by assigning a junior-level team follower. Therefore, a team member’s decision power that arises from his or her personal attributes can have the same effect on the team’s decision making

502 Decision Making in Cross-Functional Teams

Figure 4: Structure of a game in which the team follower has information and initiation power.

Figure 5: Structure of a game in which the team follower has veto power.

as his or her preferences. In cross-functional teams, a representative from each division or function will voice the concern of that function on its behalf (Griffin & Hauser, 1996). When changing the preference of one division is not desirable or feasible for a firm, adjusting the personal attributes of the representative from that division may hence be a viable alternative.

The Role of Decision Power that Arises from Participation in the Decision-Making Process

Lemma 1 shows that when the discrepancy between team members’ preferences or team members’ outside option is at a medium level such that both Conditions 1 and 2 hold, there exists a range of feasible decision outcomes that both parties prefer to pursuing their outside options. In this case, the cross-functional team’s decision outcome depends on detailed interactions between the team leader and the team follower. In this section, we exercise the framework of Mintzberg (1979) and consider three scenarios: (1) the team follower is completely excluded from the decision-making process; (2) the team follower possesses advice and initiation power; and (3) the team follower possesses veto power. Both Conditions 1 and 2 hold in the following analysis.

The first scenario constitutes the basic case of our analysis, and we model this case as the team leader’s dictator game. In the second scenario, we allow the team follower to send a signal containing his or her preference before the team leader makes the decision. In the third scenario, we model the strategic interaction

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between the team leader and the team follower as an ultimatum game in which the team leader is able to offer a “take-it-or-leave-it” proposal to the follower and the team follower’s only option is to decline or reject the proposal.

In general, any detailed bargaining between two parties can be simplified to a reduced form of bargaining in which party A (B) makes a take-it-or-leave-it offer to party B (A) of probability pA (pB ) and pA + pB = 1 (Muthoo, 2002). In the third scenario, we let pB = 0 and pA = 1 because the team follower is empowered only to veto. Therefore, the single-stage models we used in this study are the simplest representations of a bargaining model. Obviously, a veto in this context does not necessarily imply that the cross-functional team is terminated or abandoned; rather, a variety of reconciliation mechanisms may be triggered. Regardless of the details of the mechanism, such a process is not pleasant for any of the parties involved: it implies a loss of utility that both parties would prefer to avoid. In addition to the above three scenarios, we study a more intricate setting in which the supervisor can interfere and overrule the cross-functional team’s micro decision outcome on behalf of the firm to ensure alignment with its interests.

Base Case: Team Follower with No Power

We begin by analyzing a decision-making process in which the team leader (party A) has all the power and the party B has no power related to information, advice, choice, or veto. This extreme case not only establishes a baseline for comparison but also reflects the traditional practice in many contexts before the widespread adoption of cross-functional teams. By implementing this approach in the context of NPD, for example, design engineers develop a product concept without inter- ference from marketers, who simply accept design engineers’ decisions and then commercialize the product as effectively as they can. This approach to decision- making was commonly adopted in many companies until the early 1990s, and it is often called the “over-the-wall” approach, in which one party has no opportunity to express an opinion and its official mission is to work with whatever comes over the wall (Adler, 1995).

As mentioned above, we model this case as party A’s dictatorship game in which the strategic interaction between parties entails one team member’s process of optimizing his or her own interest. It quickly follows that party A chooses the decision value f solely based on his or her personal preferences (f ∗

A ) without

considering party B’s preferences. Obviously, f ∗ A

is optimal for neither party B nor the firm, and the magnitude of misalignment is |f ∗

A − f ∗|. Because there is no

communication channel from party B to party A, the team follower has no way to inform the team leader of the decision value that he or she perceives to be vital.

Team Follower with Information and Initiation Power

We now consider a decision-making process in which party B has information and initiation power. Compared with the basic case in which party B is completely excluded from the decision-making process, party B is now able to send signals to party A before party A makes a decision about value f. Thus, via communicating his or her perceived “optimal” decision value to party A, party B could potentially influence party A to make decisions that are more consistent with B’s interests.

504 Decision Making in Cross-Functional Teams

Define party B’s message space mB � R + and party A’s strategy δA with the

realization fA . Formally,

Proposition 1: For party B, a perfect Bayesian equilibrium (PBE) exists such that mB = f ∗B . For party A, δA = fA∗ for any PBE.

The proof of the above proposition is intuitive and straightforward: party B’s ability to implement information and initiation power has no material impact on party A’s utility. Consequently, irrespective of the message that party A receives, party A does not need to change behaviors and would still incorporate his or her own preferences fA

∗ into the team decision without considering party B’s opinion. The magnitude of misalignment in this case remains |fA

∗ − f ∗|. Note that in our simple formulation, the tension between both parties focuses

on the R&D budget allocation, which is then transformed into a critical single- dimensional decision value. Implicitly, the decision value in our study is cost- sensitive and thus related to the interests of both parties. Proposition 1 shows that party A insists on making decisions reflecting his or her own preferences when they have an impact on his or her well-being. However, for the decisions that are “neutral” to the cost, for example, product color in NPD context, we can also show that the party (say, party B) that cares about a cost-insensitive decision can truthfully relay his or her needs to another party (say, party A) whose own preferences are not directly affected by this decision. Party A in turn has a positive chance to be willing to consider party B’s suggestions. In this case, party A’s utility function remains identical to Equation (1), and party B and firm’s utility functions can be reformulated as follows:∏

B

= P − a ( f − f ∗B

)2 − b (ζ − ζ ∗B )2 . (4) ∏ F

= P − a ( f − f ∗

)2 − b (ζ − ζ ∗)2 . (5) where ζ denotes the aggregated value of decisions that are irrelevant to the cost, b � R and ζ � R+.

Define party B’s message space regarding the decisions that are irrelevant to the cost mζ � R

+ and party A’s strategy (δ1 A

, δ2 A

) with the realization (fA 1, fA

2), where fA

1 (fA 2) denotes the parameter for the decisions that are relevant (irrelevant)

to the cost. Following the same steps of the analysis for Proposition 1, we can show that a message mζ exists such that in PBE, ζ

∗= ζ ∗ B

, δ1 A

= fA∗ and δ2A = ζ ∗B . We do not formally incorporate this result as a proposition because it is a straightforward extension of the existing model. The general proof of this claim is available upon request.

Prior research in concurrent engineering has highlighted the benefits of communication engendered by cross-functional teams (Loch & Terwiesch, 1998; Terwiesch et al., 2002). Our analysis confirms that empowered with information and initiation power, the team follower can induce the team leader to improve the aspects of a decision that are undisputed. However, the analysis also reveals that granting the team follower information and initiation power does not address con- flicts of interest among different parties or reduce the magnitude of misalignment.

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Thus, cross-functional teams that exclusively focus on communication and infor- mation exchange may fall short of expectations when significant changes in critical tradeoffs, such as hardware vs. software, are required.

Team Follower with Veto Power

We next consider the case in which the team follower (party B) has veto power. In this case, we allow party B to object to any decision that party A may choose. A rejection by party B triggers an organizational process to resolve the conflict. For now, we do not limit the model to the specifics of a particular reconciliation process and instead seek to understand the consequences of providing party B with veto power on a more general level. (In the next section, we study in detail an exemplary hierarchical process of reconciling conflict.)

Suppose that team members are uncertain about each other’s preferences (f ∗ i

where i � {A, B}). To derive a simple but managerially intuitive result, assume that party A and party B’s preferences can be two values: either H or L. Let f H ∗

i >

f L∗ i

. Consistent with the notations defined before �t B

= {f : �B (f t∗) ≥ uB}. At the outset, party A has a subjective probability q0 that party B has a high preference f H ∗

B .

Then, party A makes an offer, f, to maximize

max

⎧⎨ ⎩

�A (f ) if f ∈ �hB q0�A (f ) + (1 − q0) uA if f ∈ �lB − �hB uA if f /∈ �lB

Under Conditions 1 and 2, party A’s proposal f ∗should be either f L B

or f H B

(anything in between would be inefficient because it would force party B, which favors low value f L∗

B , to reject the proposal without maximizing the expected payoff

to party A). These considerations lead to the following equilibrium behavior:

Proposition 2: Define the threshold probability of party At with the optimal deci- sion value fA

t∗ (where t � {H, L}) as

q t 0 =

�A ( f L

B , fA

t∗) − uA �A

( f H

B , fA

t∗) − uA . (6) Then, party At with the optimal decision value fA

t∗ imposes the following ultimatum:

f = {

f H B

if q0 > q t 0 then the probability of rejection by party B is (1 − q0)

f H B

if q0 ≤ qt0 then party B will accept. The intuition that underlies this proposition is that if party A proposes the

more aggressive decision f H B

, then this proposal will be accepted by party B with the preference f H ∗

B but rejected by party B with the preference f L∗

B . Party A’s

expected payoff then becomes q0 �A(f H B

, fA t∗) + (1 − q0 )uA. It follows that party

A would aggressively impose f H ∗ B

if and only if party A has a strong belief that party B has a preference close to his or her own, i.e., q0 �A(f

H B

, fA t∗) + (1 − q0 )uA ≥

�A(f L B

, fA t∗). In addition, party A’s threshold probability of acting aggressively

depends on his or her preference fA t∗. If party A has a preference of low decision

506 Decision Making in Cross-Functional Teams

value fA L∗, then the threshold probability, qL0 , will be higher (reflecting a greater

willingness to make a utility compromise) than it will be when his or her preference is high.

One may argue that because the team follower’s veto power creates a real risk of triggering a conflict-resolution process, which has negative consequences for both parties, allowing the team follower to signal his or her preference before the team leader submits the proposal may incentivize the team leader to behave less aggressively. In Appendix B, we consider this possibility as an extension in which the team follower is empowered with both information/initiation power and veto power. In addition, the new game relaxes the assumption that party B can only have two types of preferences. In the equilibrium of the new game, the decision outcome remains unchanged from the case in which the team follower has only veto power.

Proposition 2 therefore demonstrates that veto power for a team follower can, in a very general setting, change the decision dynamics of a cross-functional team. Unlike in the information and initiation game, in the veto game, the team follower can influence crucial decisions in his or her favor, and the team leader must refrain from radical self-optimization fA

∗. Under all circumstances, the team decision will be less preferred than that it would be by a radically self-optimizing team leader. Thus, the magnitude of misalignment changes into | f i

B − f ∗|. However, note that

the prerequisite for such a change is that the team follower has the organization’s backing in a potential power struggle (Condition 2 implies that veto power matters only with sufficiently high uB). In contrast, if the support is merely “lip service” and would not prevail during an actual conflict (low uB), then the team follower will not veto and will instead conform to the team leader’s wishes.

Proposition 2 also implies that even the faint possibility of a meaningful struggle affects team decisions. Interestingly, changes in party A’s behavior from that in the information and initiation game may occur even in the absence of an explicit veto by party B. In fact, with probability qt0, party A will choose a decision value that is always accepted by party B. Last, we should note that a drawback also arises from empowering party B with veto power - when q0 > q

t 0, the decision

value chosen by party A will indeed be vetoed by party B with probability 1 − q0. In this case, the team must endure organizational struggles, sometimes resulting in abandonment. Therefore, the cross-functional team requires the involvement of senior management to mediate conflicts.

Involving Senior Management in Micro Decision Making

In most organizations, if two parties are unable to resolve a conflict, then they can escalate decision making to a “boss” who listens to each of the parties and makes a decision. Thus, struggles between team members can usually be referred to a supervising body for resolution. Once involved in conflict resolution, the supervising entity - whether an individual or a group - makes final decisions on behalf of the cross-functional team, which has obligation to execute those decisions.

For simplicity, in modeling the hierarchy game, we assume that party A and B have one common supervisor. When party B vetoes party A’s initial decision f, then the supervisor must make the final decision. If the supervisor decides to

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Figure 6: Structure of a game in which a supervisor becomes involved to resolve conflict.

Figure 7: Supervisor’s preference and coordination cost.

overrule the team leader’s decision by executing its own will fS , a coordination cost e will be incurred, and both parties A and B must implement the supervisor’s decision. If the supervisor decides not to overrule party A’s decision, party A’s initial proposal f becomes “official” by the supervisor and will be executed, and no coordination cost e will be incurred. In organization theory, coordination costs can denote negative utility to the organizational stability or organizational efforts required to earn support from stakeholders, mobilize resources, coordinate actions and implement decisions (Gulati & Singh, 1998).

Assume that the supervisor’s utility function us (f ) is identical to the firm’s utility. Before the game shown in Figure 6 starts, the firm formulates its “vision” and informs it to its representative (supervisor), i.e., f ∗

S = f ∗and

fS ∈ R+0 .Clearly,f ∗S �= fA∗ �= f ∗B . Define �S = {f : �S (f ) ≥ �S (f ∗S ) − e }, which is the set of decision values that the supervisor prefers to implement but incur coordination cost e. Define f e

S (f e,

S as the right (left) endpoint of �S (i.e.,

�S (f e S

) = �S (f ∗S ) − e ). Figure 7 plots the supervisor’s preference and the coordination cost. From Figure 7, we can see that when e = 0, f e

S = f ∗

S . In

addition, set �S will increase with coordination cost e, which implies that as

508 Decision Making in Cross-Functional Teams

the effort required for the supervisor to interfere with the team’s micro decision increases, the supervisor’s inclination to overrule the decision made by the team leader decreases. Define ê such that f ê

S = fA∗. After the different situations are

analyzed, one general conclusion follows:

Proposition 3: Team Decision in the Supervisor Game

i. When e < ê, all PBEs yield the same decision value f e S

. ii. Otherwise, all PBEs yield the same decision value fA

∗.

In contrast, when the coordination cost becomes too high such that fA ∗ ∈ �S ,

party A can aggressively propose fA ∗ and the supervisor will not overrule no

matter how party B behaves. This is because overruling the team leader’s decision will bring less value to the firm than it costs (e). The supervisor’s decision of not overruling therefore “officializes” party A’s decision that would not have been implemented if the supervisor is not involved. In any case, the supervisor (firm) must make a tradeoff between implementing the “optimal” decision and maintaining the firm’s organizational stability.

We can quickly see from Proposition 3 that strategic misalignment (|f e

S − f ∗|) can be eliminated when the coordination cost is negligible (e � 0,

such that f e S

� f ∗ S

= f ∗). Recall that in the veto game that does not have the supervisor involved, the magnitude of misalignment is |f t

B − f ∗|. This implies

that even when coordination costs become slightly higher such that f ∗ B

<

f e S

< f t B

< fA ∗, having the supervisor involved will generate better strategic align-

ment than not having the supervisor involved. Interestingly, when the coordination cost becomes too high such that f e

S > f t

B , even the threat to involve the supervisor

will worsen the situation. Therefore, our results show that the decision to involve senior management in micro decision making should depend on characteristics that are specific to an organization, such as coordination costs.

These analytical findings echo our observations from the field study of Nokia. In the 1990s, Nokia’s CEO was determined to let Nokia gradually exit the consumer electronics market and focus solely on the telecommunication sector. As a senior engineer of Nokia recalled: “At that time, our old CEO aggressively pushed Nokia to the direction he desired ... He involved himself in each important decision and almost all budget or product development plans need his final word ... This method worked very well because he got full support from the board.” However, since 2007, Nokia’s strategic transition to the smartphone market was not successful. “We actually hired a visionary CEO who knew where we should go...Unfortunately, the old method does not work any longer because Nokia becomes much bigger than it was in 1990s. If he (the new CEO) disapproves any decision made by mid-level managers, he first needs to spend a lot of time to convince the board and get support. And very often, the board will say ‘No’ as they do not want to lose what they already have,” as the interviewee commented.

Summary of the Findings

Figure 8 summarizes our findings. First, when the preferences of team members are aligned well (f ∗

i ∈ �t

j or Condition 1 is violated), the team leader can always

implement his or her will without running the risk of rejection from the team

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Figure 8: Summary of the findings; assuming that the team leader (follower) is party A (party B) with preference fA

∗(f ∗ B

) and outside option uA (uB), the supervisor bears coordination cost e, t � {H,L}.

follower (Case A). In contrast, with excessive preference discrepancy in the team (�i (f

t j ) < ui or Condition 2 is violated), the team members will lose the incentive

to stay, and the team will likely collapse (Case F). When the discrepancy between preferences is at a medium level (both Conditions 1 and 2 hold), how team members participate in the decision-making process will determine the team decision (Case B, C, D, E).

Second, team members’ decision power that arises from their personal at- tributes may play a role similar to the decision power that arises from their prefer- ences. However, it could be used an independent lever to counteract the undesired decision outcome when changing team members’ preferences is not feasible. For example, in Case A, in which Condition 1 is violated, assigning a senior team follower with a high outside option will make Condition 1 hold (f ∗

i /∈ �t

j ). This

scenario may generate the desired conflict to mitigate strategic misalignment. In contrast, in Case F, assigning a junior team follower with a low outside option can make Condition 2 hold (�i (f

t j ) ≥ ui ). This scenario will incentivize the team

follower to remain on the team. Third, the baseline model demonstrates that when the team follower is ex-

cluded from the decision-making process, the team decision tends to reflect only the team leader’s interest (Case B). This case yields the same magnitude of strate- gic misalignment as Case A. Granting information and initiation power to the team follower may alter a cross-functional team’s decision outcome with respect to aspects that are not vital to the team leader’s interests. However, granting the

510 Decision Making in Cross-Functional Teams

team follower such power alone does not affect the organization’s power structure because the team’s decision outcome may still be prone to misalignment with the firm’s objectives regarding the aspects that are vital to the interests of the team leader (Case C). Remarkably, granting the team follower veto power may cause a noticeable shift in both the team’s decision outcome and the magnitude of mis- alignment (Case D). Finally, a frequently used method of substantiating veto power is to involve the senior management in conflict resolution. By making appropriate decisions in the event of a veto, a supervisor can support or weaken a particular position, depending on the cost of coordination (Case E).

CONCLUSION AND DISCUSSIONS

Through a series of game theoretical models, this study models a joint decision- making process in cross-functional teams. It shows three factors will jointly deter- mine the outcome of the team decision: the discrepancy between team members’ preferences; team members’ decision power that arises from their personal at- tributes; and team members’ decision power that arises from participation in the decision-making process. Our results have implications for both research and practice.

Implications for Research

From the perspective of aligning a firm’s macro objective with a cross-functional team’s micro decision, this study contributes new insights regarding the effi- cacy of some well-established project management principles in the literature on cross-functional teams (Griffin, 1997; Griffin & Hauser, 1996). For example, prior research highlights the importance of adjusting team composition by involv- ing “right” functionalities in the team (Henke et al., 1993; Natter et al., 2001). Instead, our study shows that firms should also carefully manage team compo- sition in terms of team members’ personal attributes such as seniority or status in organizations, which influence the team member’s decision power and the de- cision outcome. Communication is another “success driver” often mentioned in the cross-functional team literature (Henke et al., 1993; McDonough, 2000). Our study shows that empowering the team member with information and initiation power can indeed improve some aspects of decision making. However, to make a critical change in a team’s decision making, facilitating communication by em- powering only one party with information/initiation power is not sufficient; it has to be backed up by veto power. In addition, our study provides a contingent view regarding “team autonomy” (McDonough, 2000). Different from the prescription of prior research, our study suggests that firms should maintain team autonomy and refrain from interfering with team decisions only when the team leader’s preference is aligned with firm’s preference or the coordination cost is too high.

To the best of our knowledge, this study is one of the few studies in the literature on NPD that examine the horizontal distribution of decision powers among team members as opposed to the vertical distribution of decision power (Hutchison-Krupat & Kavadias, 2014). Exercising Minzberg’s framework, we de- fine four types of decision power that arise from team members’ participation in

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Figure 9: A managerial framework in aligning a team’s decision with the firm’s objective.

the decision-making process (information, initiation, choice and veto) and build several analytical models to represent different situations of decision making. The four types of decision power in our study are quite generic. We do not intend to provide an exhaustive “list” of decision powers - obviously, any such “list” will depend on the specific context of decision making.

In taking a wider perspective on implementing any firm’s level of initiative, the academic and practitioner literature has often implicitly assumed that decision makers will make the “right” decision when they are presented with the “right” information (Griffin & Hauser, 1996; Terwiesch et al., 2002). The models that we have developed in this study support the view that increasing the availability of in- formation may improve some aspects of decision making (i.e., those not vital to the decision maker’s interests). However, to adopt a holistic and strategic perspective on decision making, the value of information sharing must be complemented with an understanding of how team members’ preferences and decision power shape conflict resolution and dictate choices involving different tradeoffs in decision making.

Implications for Practice

Figure 9 summarizes a managerial framework for practitioners. It shows that to facilitate the alignment between a team’s micro decision and the firm’s objective, the choice of management levers should consider (1) which party’s preference is more “aligned” with the firm’s objective, indicating the “right” direction to pursue and (2) the discrepancy between the team members’ preferences. Fixing one and slightly changing another will significantly change the management’s prescriptions.

In the following discussions, for the ease of notation, definef̃ = max{f t

B , f e

S }. Let’s begin by considering a scenario in which the firm’s objec-

tive (f ∗) is closer to the team follower’s preference (f ∗< f̃ +fA ∗2 ). It echoes the Nokia

512 Decision Making in Cross-Functional Teams

Figure 10: Structure of a game in which the team follower has both initiation power and veto power.

example, where the hardware-driven team leader was reluctant to switch the focus of product development to software and services. In this case, if the preferences of the team members are perfectly aligned (Condition 1 is violated), the magnitude of strategic misalignment will be maximized (|fA∗ − f ∗|). Firms could consider changing the team leader or assigning a senior-level team follower to balance the influence of the team leader if changing the leader is not a feasible option. How- ever, when the discrepancy between the team members’ preferences is so high that Condition 2 is violated, no joint decision can be made by the team. In this extreme case, firms could consider organizing training sessions or job rotations to reduce the discrepancy of preferences between team members (e.g., Griffin and Hauser, 1996) or adjusting the team composition by assigning a junior-level team leader. Both levers can make Condition 2 hold and reduce the misalignment (|f t

B − f ∗|).

When the discrepancy between the team members’ preferences is medium such that both Conditions 1 and 2 hold, assigning a senior team follower (high uB) or empowering the team follower with veto power can shift the team decision. Last, introducing the threat of overruling team decisions by top management may also shift team decision outcomes toward the direction that will benefit the firm (|f e

S − f ∗|). Conversely, when the firm’s preference is more closely aligned with the team

leader’s preference (f ∗ > f̃ +fA ∗

2 ), it is in the firm’s best interest to maximize the

team leader’s influences on the team decision, and all of the levers discussed above should be triggered in the opposite direction. For example, instead of changing the team leader, the firm should rather maintain the status quo when the discrep- ancy of the preferences in the team is very low. In an extreme case in which the firm’s and the team leader’s preferences are perfectly aligned (fA

∗ = f ∗), the team follower should be excluded from the decision-making process because it elimi- nates potential noise. Despite being a rare situation in reality, for completeness, when f ∗ = f̃ +fA ∗

2 , the firm is indifferent between maintaining the status quo and

strengthening the team follower’s decision power.iii

iii In this case, |f̃ − f ∗| = |fA∗ − f ∗| and the firm does not reduce misalignment by empowering the team follower.

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The managerial framework provided in our study allows a firm to system- atically influence its teams’ decisions even when it does not fully understand the detailed technical aspects of those decisions. However, our study also shows that each lever has both advantages and disadvantages for a firm’s strategic alignment depending on the organizational context, such as the discrepancy between the preferences and the coordination costs. Thus, no single lever is suitable for all or- ganizational contexts; rather, an “appropriate” lever exists for a given firm-specific initiative.

Example: Apply Decision Power as a Lever to Facilitate Alignment

Let us go back to the development of the BMW 7-series sedan (Pisano, 2002) to illustrate how to apply decision power as a lever to align a cross-functional team’s micro decision with the firm’s objective. The levers studied in this paper could help BMW reposition its product from being “performance quality-focused” to being “conformance quality-focused.”

BMW could first consider aligning the preferences of its team members by using traditional project management methods such as organizing training ses- sions, facilitating communication, and creating job rotations. In particular, design engineers should be trained to accept top management’s vision that BMW needs to be repositioned in the direction of “conformance quality” and “cost reduction.” In addition, BMW should also take into account team members’ decision power that arises from their personal attributes. For example, to balance the dominant influence exerted by a design engineer, BMW could assign a senior manufac- turing engineer (i.e., a team follower) with strong personal attributes relative to design engineers (a team leader). Moreover, BMW could consider restructuring the decision-making process and empowering manufacturing engineers with the power to veto the design. For example, a manufacturing engineer could possibly veto every design proposal that is unable to meet the requirements of confor- mance quality or target manufacturing costs. However, we do not suggest that BMW should directly involve senior management in micro product development decisions because in a large-scale complex organization such as BMW, if top man- agement breaks the hierarchy and micromanages a team’s decision making, such actions are often accompanied by a significant amount of negative utility, such as internal conflicts, a loss of loyalty, etc. In this case, the threat of overruling a team leader’s micro decision is not truly credible because doing so will generate more costs than gains for a firm.

Discussions and Limitations

Clearly, decision power is only one of several mechanisms firms can use to steer cross-functional teams’ micro decision making. As a well-established alternative, firms could introduce and construct monetary incentives (e.g., Mihm, 2010). In our study, firms could also use revenue-sharing contracts, i.e., team members receive a portion of the revenue generated by the project such that the firm’s and

514 Decision Making in Cross-Functional Teams

team’s incentives are better aligned. In addition, firms could consider imposing penalties (awards) if the team decision deviates from (conforms with) the firm’s preferences. However, there could be two major limitations regarding the use of monetary incentives. First, monetary incentives tend to be costly; at a minimum, they require that (at least) one agent be taxed (i.e., the one who already acts in line with the firm’s interests) so that another agent can be subsidized (i.e., the one who acted against the firm’s interests in the first place) to act more in line with the firm’s interests (Mihm, 2010). However, such a setup may become implausible for most organizations due to considerations of fairness. Second, the implementation of these mechanisms requires the firm to have a fully specified utility function, a substantial amount of information regarding the agent (team member)’s utility functions as well as good knowledge of the agent’s actions. These requirements are often unrealistic in practice, particularly in the fuzzy and uncertain decision- making context (Ulrich & Eppinger, 2003).

This study considers a cross-functional team composed only of two critical parties, and a supervisor may intervene as a mode of conflict resolution. In practice, a large-scale cross-functional team can sometimes be composed of multiple parties who all play important roles in a team’s decision making. In that case, team members’ participation in decision-making processes will be quite complex, and the stylized models in our study may need to be significantly extended to capture more subtle insights. How to distribute decision power horizontally among multiple team members is a challenging yet promising research question to be explored in the future.

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Appendix A

SUMMARY OF ASSUMPTIONS AND PARAMETERS

Assumption 1.: Single Dimensional decision space f and f ≥ 0 Assumption 2.: Quadratic utility functions for the firm and team members

Cui 517

Assumption 3.: f ∗ �= fA∗ = f ∗B Parameters

f Decision value

P The maximum payoff potential from successfully making the decision

ui Party i’s outside option

�i Party i’s expected payoff

f ∗ i

Each party’s perceived “optimal” decision value that maximizes its expected payoff

�t i

The set of decision values party i with preference f ∗ i

prefers to receive its outside option

f t B

The right endpoint of �t B

. Formally, it is a value from �t B

maximizing party A’s expected utility

e The supervisor’s coordination cost

�S The set of decision values the supervisor prefers to implement but incur coordination cost e

f e S

The right endpoint of �S . Formally, it is a value from �S maximizing party A’s expected utility.

Appendix B

MODEL EXTENSION: TEAM FOLLOWER WITH BOTH INFORMATION/INITIATION AND VETO POWER

In this section, we consider the possibility that the team follower is empowered with both information/initiation and veto power. The team follower is able to choose and send (after observing his or her optimal decision value) a message m from a large but finite set of messages, M. After sending the message, the game proceeds exactly as a veto with identical payoffs. Formally, let nature choose party B (the team follower)’s type (perceived optimal decision value f ∗

B ) from the

set T = {f 0∗ B

, f 1∗ B

, f 2∗ B

· · · , f n∗ B

}, n � 0, according to a discrete prior distribution h(.), in which h(f i∗

B ) = P r(f ∗

B = f i∗

B ) and h(f i∗

B ) > 0 for all f i∗

B ∈ T . The element

of T satisfies f i∗ B

< f j ∗ B

for all i < j ,where i, j ∈ N ≡ {0, 1, 2, · · · , n}. Define party B to adopt a message strategy that gives a probability distribu-

tion for all possible messages in M for each type in the set T. Let p(m, f i∗ B

) be the probability that type f i∗

B sends message m in a given equilibrium. The party A (the

team leader)’s decision strategy now associates with each m a probability distribu- tion over [0, 1]. Let f (m) be the decision made by party A on hearing m whenever party A does not mix given the announcement. In any given equilibrium, party A will form posterior beliefs about party B’s type for each message m. Let these be denoted h(f i∗

B , m) ≡ P r(f ∗

B = f i∗

B |m) with H (f i∗

B , m) ≡ ∑i−1

j =0 h(f j ∗ B

, m). Let

H (f i∗ B

) ≡ ∑i−1 j =0 h(f

i∗ B

, m) denote the prior probability that party B’s optimal

518 Decision Making in Cross-Functional Teams

decision value is strictly less than f i∗ B

. Let H (f 0∗ B

) = 0. For simplicity of anal- ysis, we assume that if party B is indifferent between rejecting and accepting a request f, then party B definitely accepts the request.

Proposition 4: If f̂ ∗ is the unique equilibrium decision value in the veto game, then in any equilibrium of the game in which party B has information/initiation power and in which party A uses a pure strategy, party A requests f̂ ∗ regardless of the received message.

To better understand this result, suppose that party A conditioned his or her behavior on f, seizing more or less depending on what party B communicated. Then, regardless of the true willingness to receive fallback utility, party B wants to communicate his or her preference in a way that leads to the smallest seizure by party A - that is, party B has an incentive to misrepresent his or her actual preference of optimal decision value. Then, however, party A learns nothing from the oral communication. In sum, although team members have an incentive to avoid receiving an outside option, they also wish to obtain a favorable decision outcome. This desire may motivate them to exaggerate their true preferences, if by doing so they can persuade the other side to make concessions. Similarly, team members may also have an incentive to conceal their true outside options if they are concerned that revelation would cause them to “look weak.”

Appendix C

PROPOSITIONS AND PROOFS

Proof of Lemma 1: It is sufficient to show that an interval [fA

t , f t

B ] ⊆[0,f̄ ] is non-empty. Suppose that

fA t > f t

B . Note that fA

t is the value such that P − (fAt − fAt∗)2 = uA and fAt < fAt∗ . It follows that when fA

t > f t

B , P − (f t

B − fAt∗)2 < uA which contradicts with

condition 2 (common ground assumption). �

Proof of Proposition 1: Prove by contradiction. If in some equilibrium, party A chooses any f ′ �= fA∗ following a message m then party A receives �(f ′), but then party A could deviate to f = fA∗ and receive �(fA∗). By definition of fA∗, �(fA∗) > �(f ′) and party A is strictly better off. �

Proof of Proposition 2: Let f ∗ denote the optimal decision value party A requests. f ∗ needs to maximize �A(f ) subject to f ∈ �tB . By Conditions 1 and 2, f ∗ should be either f HB or f L

B . If party A proposes f L

B , it will be accepted by party B of any type and party

A’s expected payoff is �A(f L B

). If party A proposes f H B

, it will be accepted by a high-type party B and rejected by a low-type party B. Party A’s expected payoff is q0�A(f

H B

) + (1 − q0)uA. Therefore, party A of type t would propose f HB if and only if qt0�A(f

H B

, fA t∗) + (1 − qt0)uA ≥ �A(f LB , fAt∗), or qt0 ≥

�A(f L B

,fA t∗)−uA

�A(f H B

,fA t∗)−uA .

Cui 519

Observe that �A(f

L B

,fA t∗)−uA

�A(f H B

,fA t∗)−uA is decreasing infA

t∗. Since fA t∗ ∈ {fAH ∗, fAL∗}

and fA H ∗

> fA L∗, we can therefore define two threshold variables critical for the

following analysis:

q H 0 =

�A ( f L

B , fA

H ∗) − uA �A

( f H

B , fA

H ∗) − uA qL0 = �A

( f L

B , fA

L∗) − uA �A

( f H

B , fA

L∗) − uA It follows that when q0 < q

H 0 , party A would request f

L B

. When q0 > q L 0 ,

party A would request f H B

. When qH0 < q0 ≤ qL0 , party A of optimal decision value fA

H ∗requests f H B

, and of fA L∗imposes f L

B . �

Proof of Proposition 3: The game involving the supervisor can be modeled as a three-stage game: at the first stage, party A propose f ; at the second stage, party B decides whether to accept; at the third stage, if party B accepts f, the supervisor decides f as well and if party B rejects, the supervisor decides f ′ and incurs coordination cost e (the sequence of events is shown in Figure 6). �

Formally, let at i ∈ Ai (ht ) denote player i’s stage-t action, where Ai (ht ) is the

action set player i faces at stage t (t � {1, 2, 3}). Define history at the beginning of stage t as ht = (a0, · · · , at−1). We assume that party A and the supervisor don’t know party B’s type f t∗

B but share a common belief on party B’s type. Party B’s

perceived optimal decision value f t∗ B

is distributed in the same way as described in Appendix B.

We further define the action sets, strategy and belief of relevant players at each stage.

At Stage One, define that party A’s action a1 A

= f ∈ R+, supervisor a1 S

∈ ∅. Party A’s strategy σA is defined as

σA ( a

1 A

) ∈ arg max

a1 A

Eσ 3s

[ Eσ 2

B

[ Ef ∗t

B [�A (f )]

]] . (C1)

where f ∈ { a 1 A

if a2 B

= 0 (accept) a3

S if a2

B = 1 (reject) .

At Stage Two, define party A a1 A

∈ ∅ and supervisor a2 S

∈ ∅. Party B’s action a2

B = aB (a1A) ∈ [0, 1], defined as a mixed strategy to reject the proposal from party

A. Party B’s mixed equilibrium strategy σ 2 B

is the support of action a2 B

. Hence at stage two, for any equilibrium strategy σ 2

B and the alternative

strategy σ 2 ′

B , it must be true that:

Eσ 3s

[ aB2 �B (σS3 , ·) + (1 − aB2 ) �B

( a

1 A, ·

)] ≥ Eσ 3s

[ a

2′ B �B

( σ

3 S , ·

) +

( 1 − aB2′

) �B (aA1 , ·)

] Cancel the common items on both sides and reorganize the equations, we get

a 2 B Eσ 3S

[ �m

( σ

3 S , ·

) − �m

( a

1 A, ·

)] ≥ a2′B Eσ 3S

[ �m

( σ

3 S , ·

) − �m

( a

1 A, ·

)]

520 Decision Making in Cross-Functional Teams

The strategy of σ 2 B

can therefore be defined as:

∀ f t∗B , σ 2B ∈ arg max a2

B

a 2 B Eσ 3S

[ �m

( σ

3 S , ·

) − �m

( a

1 A, ·

)] . (C2)

At Stage Three, define the action set of party A and party B as a3 A

∈ ∅ and a3

A ∈ ∅ respectively. The supervisor’s action set a3

S (a1

A , a2

B = 1) = R+ and

a3 S (a1

A , a2

B = 0) = a1

A .

The strategy of supervisor σ 3 S

is therefore defined as:

σ 3 S ∈ arg max

a3 S

Ef t∗ B

[ �s

( a

3 S

)] Since the supervisor’s payoff does not depend on the preference of party B,

we can simplify it as:

σ 3 S ∈ arg max

a3 S

�s ( a

3 S

) . (C3)

The supervisor’s belief is updated by the Bayesian rule:

μS ( f

t∗ B |a2B

) = h

( f t∗

B

) σB

( a2

B |f t∗

B

) ∑h (

f t∗ B

) σB

( a2

B |f t∗

B

) . (C4) Definition 1. In this game, a perfect Bayesian equilibrium is a profile (σ 1

A ,

σ 2 B

, σ 3 S

, μS ) that satisfies (C1)–(10). For simplicity, we assume party B always plays a pure strategy, i.e., a2

B =

1 or a2 B

= 0. The proof of the first part of Proposition 3 will be sufficed to show that when e < ê, profile (σ 1

A , σ 2

B , σ 3

S , μS ) with a

1 A

= f e S

, a2 B

= 1or 0, a3 S

= f e S

are unique Perfect Bayesian Equilibriums. We will show the existence and then the uniqueness by excluding other potential equilibriums.

Given party B and the supervisor’s strategy a2 B

= 1, a3 S

= f e S

, by Equation (C1), party A’s expected payoff is �A(f

e S

). If party A deviates by proposing any value a1

A = f ′ > f e

S , a2

B = 1 and the supervisor will overrule and implements f ∗ as

�S (f ∗) − e > �S (f ′)by definition of f eS . The expected payoff of party A becomes

�A(f ∗) and party A gets strictly worse off. If party A deviates by proposing any

value a1 A

= f ′ < f e S

, the supervisor will not overrule and the expected payoff of party A is �A(f

′). Party A can get strictly better off by increasing f ′ to f e S

. Given a1

A = f e

S , σ 3

S = f e

S , �B (a

3 S , ·) − �B (a1A, ·) = 0, so σ 2B = a2B = 1 full-

fills Equation (C2). And party B is indifferent between a2 B

= 1 and a2 B

= 0. Given a1

A = f e

S , σ 2

B = 1, by definition of f e

S , σ 3

S = f e

S fulfills the Equation

(C3) as e < ê. Hence, profile (σ 1

A = a1

A = f e

S , σ 2

B = 1, σ 3

S = f e

S ) is a perfect Bayesian

equilibrium. Similarly, we can show that profile (σ 1 A

= a1 A

= f e S , σ 2

B = 0, σ 3

S =

f e S

) is a Perfect Bayesian Equilibrium. Next, we show that for the supervisor game, there is no Perfect Bayesian

Equilibrium such that a1 A

< f e S

. Assume to the contrary, there exists a Perfect Bayesian Equilibrium such that

a1 A

< f e S

. We now respectively consider two scenarios. Case One: f e ′

S ≤ a1

A < f e

S .

In this case, a3 S

= a1 A

no matter a2 B

= 0 or a2 B

= 1, because �S (a1A) > �S (f ∗S ) − e. Thus �A = �A(a1A). Party A gets strictly better off by setting a1A = f eS , which

Cui 521

contradicts with the assumption. Case Two: f e ′

S > a1

A . In this case, when a2

B = 0,

it follows that a3 S

= a1 A

and party A’s payoff is �A(a 1 A

) < �A(f e S

); when a2 B

= 1, a3

S = f ∗

S and party A’s payoff becomes �A(f

∗ S

) < �A(f e S

). In both cases, party A is strictly better off by setting a1

A = f e

S . Therefore a PBE in which f e

′ S

> a1 A

cannot exist. Similarly, following the same step of analysis, we can show that a PBE in which a1

A > f e

S cannot exist.

The second part of Proposition 2 directly follows from e ≥ ê and the defini- tion of fA

∗.

Proof of Proposition 4: The proof follows from several lemmas.

Lemma 2: In any equilibrium party A’s payoff is at least (2

√ P − uBfA∗ − (f ∗A2) + uB) and in no equilibrium will party A respond to any

message with a proposal that is certain to result in veto.

Proof of Lemma 2: If party A sets f = √P − uB, all types of B will accept for sure in any equilibrium (by subgame perfection), so party A can assure it- self (2

√ P − uBfA∗ − (f ∗A2) + uB)in this way. It can be easily shown that

(2 √

P − uBfA∗ − (f ∗A2) + uB) ≥ uA by Condition 2. If in some equilibrium party A chooses f following a message m such that it is certainly vetoed by party B, then party A receives uA; in that case, however, party A could deviate to f ′ = √P − uBand do strictly better. Lemma 3: In any equilibrium, the decision value f is in the support of party A’s proposal strategy given a message m only if there exists a f i∗

B ∈ T such that

f = f i∗ B

+ √P − uB. Proof of Lemma: If not, then party A might on hearing m choose a request f ′ such that f i∗

B +√

P − uB < f ′ < f i+1∗B + √

P − uB. But then party A would increase his or her payoff on hearing m by deviating to f ′′ = f i+1∗

B + √P − uB, since doing so has

no effect on the risk of being vetoed.

Lemma 4: In any equilibrium in which party A does not mix, we have f (m) = K, a constant, for all messages m ∈ F ′, where F ′ is the set of messages sent with positive probability in the given equilibrium.

Proof of Lemma: Let Tm ≡ {f i∗B : p(m, f i∗B ) > 0}. Suppose to the contrary that in some equilibrium there exist two distinct message m and m’ such that f (m) < f (m′). By Lemma 2, both requests must be accepted with positive probability, implying that there are types f i∗

B ∈ Tm such that P − (f (m) − f i∗B )2 ≥ uB and types f

j ∗ B

∈ Tm′ such that P − (f (m′) − f j ∗

B )2 ≥ uB. But then, any such party B of f j ∗B ∈ Tm′ can do better

by deviating to m, which yields P − (f (m) − f j ∗ B

)2 ≥ P − (f (m′) − f j ∗ B

)2. But this implies that f (m′) is certainly rejected, contradicting Lemma 2.

With above lemmas, we now prove the Proposition 4. Suppose f ∗ = f k∗

B + √P − uB is the unique request in the veto game without em-

powering party B with initiation/information power. Then, f k∗ B

is the only element

522 Decision Making in Cross-Functional Teams

of T such that for all j ∈ N ,

H (f k∗B )uA + [1 − H (f k∗B )][P − ( √

P − uB + f k∗B − fA∗) 2 ] ≥ H (f j ∗

B )uA

+ [1 − H (f j ∗ B

)][P − ( √

P − uB + f j ∗B − fA∗) 2 ]. (C5)

Suppose to the contrary of the proposition that in the veto game empow- ering party B with initiation/information power, there is some other requests f ′ = √P − uB + f l∗B , l �= k, such that party A requests f ′ on hearing any message m ∈ F ′ (by Lemma 4, any equilibrium without mixing by party A must have this form). Then it must be that for each m ∈ F ′ and for all j ∈ N ,

H ( f

l∗ B , m

) uA +

[ 1 − H

( f

l∗ B , m

)] [ P −

(√ P −uB +f l∗B −fA∗

)2] ≥ H

( f

j ∗ B

) uA

+ [ 1 − H

( f

j ∗ B

)] [ P −

(√ P − uB + f j ∗B − fA∗

)2] . (C6)

By Bayes’s ruleh(f j ∗ B

, m) = h(f j ∗ B

)p(m, f j ∗ B

)/P r(m) and H (f j ∗ B

, m) = (1/P r(m))

∑j −1 i=0 h(f

j ∗ B

, m), where P r(m) = ∑ f ∗

B ∈T h(f

∗ B

)p(m, f ∗ B

) is the proba- bility that party B chooses message m in the equilibrium. Substitution into Equation (C6) and multiplication of both sides by P r(m) yields, for all j ∈ N ,[

l−1∑ i=0

h(f i∗B )p(m, f i∗ B )

] uA +

[ P r(m) −

l−1∑ i=0

h(f i∗B )p(m, f i∗ B )

] [ P − (√P − uB + f l∗B − fA∗)

2 ]

≥ [

j −1∑ i=0

h(f j ∗B )p(m, f i∗ B )

] uA +

[ P r(m) −

j −1∑ i=0

h(f i∗B )p(m, f i∗ B )

] [ P − (√P − uB + f j ∗B − fA∗)

2 ] .

(13)

Since Equation (13) holds for each m ∈ F ′, we can sum both sides over all m ∈ F ′ and the inequalities still hold. Thus, for all j ∈ N ,[

l−1∑ i=0

h(f i∗ B

) ∑

m∈F ′ p(m, f i∗

B )

] uA+

[ ∑ m∈F ′

Pr(m) − l−1∑ i=0

h(f i∗ B

) ∑

m∈F ′ p(m, f i∗

B )

] [ P − (√P − uB + f l∗B − fA∗)

2 ]

≥ [

j −1∑ i=0

h(f i∗ B

) ∑

m∈F ′ p(m, f i∗

B )

] uA

+ [ ∑

m∈F ′ P r(m) −

j −1∑ i=0

h(f i∗ B

) ∑

m∈F ′ p(m, f i∗

B )

] [ P −(√P −uB + f j ∗B − fA∗)

2 ] .

(14)

Since ∑

m∈F ′ P r(m) = 1 and ∑

m∈F ′ p(m, f i∗ B

) = 1, Equation (14) simplifies to yield, for all j ∈ N ,

H ( f

l∗ B

) uA +

[ 1 − H

( f

l∗ B

)] [ P −

(√ P − uB + f l∗B − fA∗

)2] ≥ H

( f

j ∗ B

) uA

+ [ 1 − H

( f

j ∗ B

)] [ P −

(√ P − uB + f j ∗B − fA∗

)2] . (C7)

But this contradicts Equation (C5), the hypothesis that the veto game without empowering party B with initiation/information power has a unique equilibrium, f̂ ∗ = f k∗

B + √P − uB, where f k∗B

Cui 523

Dr. Zhijian Cui is an assistant professor of operations management at IE Business School and a visiting assistant professor at Hong Kong University of Science and Technology Business School. His research interest mainly focuses on the management of knowledge-intensive processes such as new product development, R&D, etc. He received his PhD in Operations Management from INSEAD, MBA from Pepperdine University, M.S. in Management from Kedge Business School and Bachelor of Engineering from Tsinghua University.

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