a)Discuss the property, prove/disprove if Nash solution satisfies property. b)Determine Nash solution for this problem and discuss
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bargaining theory (cooperative and non-cooperative). This lecture covers Nash’s original work.2
2 Nash’s Approach to the Bargaining Problem.
2.1 Nash’s Bargaining Games.
We now introduce a model of the bargaining problem due to Nash. The model is incredibly simple, abstracting from all unnecessary details. A bargaining game is simply a set 𝑆 where:
• The set 𝑆 ⊂ ℝ2 is a nonempty, compact and convex set3, called the set of alternatives. Each alternative 𝑎 ∈ 𝑆 is a two-dimensional vector 𝑎 = (𝑎1, 𝑎2) specifying a utility level 𝑎1 for player 1 and a utility level 𝑎2 for player 2.
• The set 𝑆 contains the origin (0, 0), which will be called the disagreement point. If the players fail to reach an agreement, the utilities received from disagreement are set to zero.
The following diagram illustrates a typical bargaining problem, where the set shaded blue is the bargaining game:
2This is the same John Nash who proposed the Nash equilibrium concept you will know from non- cooperative game theory. Nash’s bargaining solution is not directly related to that concept.
3A set 𝑆 ⊂ ℝ2 is compact if it is closed and bounded. So, it includes its own boundary, and can be contained in a (large enough) square. A set is convex if any two points in the set can be connected by a straight line that is contained in the set.
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Utility for player 2
Utility for player 10
S
a1
a2 a
It is always assumed that there is an alternative 𝑎 ∈ 𝑆 with 𝑎 ≫ (0, 0).
Example 1: Consider the problem of “dividing a dollar” (or any amount, of any currency). Suppose that two players, 𝐴 and 𝐵 are bargaining over a surplus of size one. Each player 𝑖 = 𝐴, 𝐵 has strictly increasing and strictly concave utility 𝑢𝑖 with 𝑢𝑖(0) = 0. The set of alternatives for this problem is then:
𝑆 = {(𝑢𝐴(𝑥), 𝑢𝐵(𝑦)) ∶ 𝑥 ⩾ 0, 𝑦 ⩾ 0, 𝑥 + 𝑦 ⩽ 1},
with the disagreement point being (0, 0). It can be confirmed that 𝑆 is a convex set using the fact the utilities are concave.
Example 2: Suppose we have an oil painting 𝑃 and that two players want to bargain over who gets it. The painting cannot be chopped up (nobody wants this), so the players must reach a decision who gets the painting. Nobody gets it if there is disagreement. Let us scale
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the von-Neumann utilities so that 𝑢𝐴(𝑃 ) = 𝑢𝐵(𝑃) = 1 and 𝑢𝐴(0) = 𝑢𝐵(0) = 0. Now, in this case, the set of possible outcomes is:
𝑋 = {(1, 0), (0, 1), (0, 0)}
and the bargaining game would seem to be:
𝑆 = {(𝑢𝐴(𝑥), 𝑢𝐵(𝑦)) ∶ (𝑥, 𝑦) ∈ 𝑋},
But, this is not a convex set. How then can we justify Nash’s assumption of convexity? The idea is simple - allow the bargainers to use lotteries. Let 𝑋 denote the set of lotteries over the set 𝑋. If the players use expected utility, the bargaining game is then:
𝑆 = {(𝑢𝐴(𝑥), 𝑢𝐵(𝑦)) ∶ (𝑥, 𝑦) ∈ 𝑋}, and 𝑆 is indeed a convex set. For example, the point (1
2 , 1 2 ) corresponds to each player
receiving a 50-50 chance of being the one to receive the painting. Using expected utility and lotteries means convexity is guaranteed. Indeed, a lottery solution is very natural here. The obvious way to solve the problem is to flip a coin.
2.2 Solution Concepts.
We now have a very general way of modelling bargaining games. In terms of determining who gets what, we need a solution concept. In noncooperative game theory, we know so- lution concepts such as competitive equilibrium, Nash equilibrium, subgame perfect equi- librium, Bayes-Nash equilibrium, and so on. The bargaining games we study are part of what is know as cooperative game theory. We take a very general point of view, and define a solution concept as a function (a rule) that selects an alternative for any bargaining game:
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Definition (Solution Concept):
A solution concept is a function 𝜑 associating each bargaining game 𝑆 with an alter- native 𝜑(𝑆) ∈ 𝑆.
If the solution concept selects alternative 𝜑(𝑆) = (𝜑1(𝑆), 𝜑2(𝑆)), then player 1 re- ceives utility 𝜑1(𝑆) and player 2 receives utility 𝜑2(𝑆).
At this point, the class of solution concepts is enormous, so we need to narrow it down. In the next section, we list four properties (axioms) due to Nash that we might like our solution concept to satisfy.
2.3 Axioms.
The first of Nash’s axioms should be familiar, requiring that the alternative selected by our solution concept is Pareto efficient:
Axiom 1 (Pareto Efficiency):
For all bargaining games 𝑆, the solution concept selects an alternative 𝜑(𝑆) that is Pareto efficient. That is, there is no 𝑎 ∈ 𝑆 with 𝑎 ⩾ 𝜑(𝑆) and 𝑎 ≠ 𝜑(𝑆).
Pareto efficiency seems to be a desirable, and minimal, property for a solution to a bargain- ing problem. Clearly, suggesting that some of the surplus being bargained over is destroyed is bad advice. Also, if an agreement was made that was Pareto dominated by another, then presumably neither player would object to changing the agreement.
Here is Nash’s second axiom:
Axiom 2 (Symmetry):
If the bargaining game 𝑆 is symmetric, then the solution concept 𝜑 should select an alternative 𝜑(𝑆) that is also symmetric. Formally, if (𝑎1, 𝑎2) ∈ 𝑆 implies (𝑎2, 𝑎1) ∈ 𝑆, and 𝜑(𝑆) = (𝜑1(𝑆), 𝜑2(𝑆)), then 𝜑1(𝑆) = 𝜑2(𝑆).
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The symmetry axiom requires that, if the bargaining problem is symmetric, then the solu- tion should be symmetric. So, two identical individuals bargaining over how to divide a surplus should divide it equally.
The third axiom is the following:
Axiom 3 (Linear Invariance):
The physical alternative selected by the solution concept should be the same for all equivalent utility representations. That is, if bargaining game 𝑆 is transformed to 𝛼𝑆, with 𝛼1 > 0, 𝛼2 > 0, then:
𝜑(𝛼𝑆) = 𝛼𝜑(𝑆).
To understand the linear invariance axiom, consider the problem of dividing £50,000 be- tween Ann and Bob. Suppose that the disagreement point is zero for both. We are fixing the utilities of the disagreement point so that 𝑢𝐴𝑛𝑛(0) = 0 and 𝑢𝐵𝑜𝑏(0) = 0. Then, the bargaining set becomes:
{(𝑢𝐴𝑛𝑛(𝑥), 𝑢𝐵𝑜𝑏(𝑦)) ∶ 𝑥 ⩾ 0, 𝑦 ⩾ 0, 𝑥 + 𝑦 ⩽ £50, 000}
Now recall that von Neumann-Morgenstern utilities are cardinal. This means Ann’s utility 𝑢𝐴𝑛𝑛 could be replaced by 𝑣𝐴𝑛𝑛 = 𝛼𝑢𝐴𝑛𝑛, with 𝛼 > 0 and 𝑣𝐴𝑛𝑛 is also a von Neumann- Morgenstern utility.4 So, we could have used a different utility representation of the prob- lem:
{(𝛼1𝑢𝐴𝑛𝑛(𝑥), 𝛼2𝑢𝐵𝑜𝑏(𝑦)) ∶ 𝑥 ⩾ 0, 𝑦 ⩾ 0, 𝑥 + 𝑦 ⩽ £50, 000}
where 𝛼1, 𝛼2 > 0. This is simply a different numerical representation – the problem has not changed! So, the linear invariance axiom says that the solution should not change either. The solution, in utility space, should be the same as before but simply multiplied by the constants 𝛼1 and 𝛼2 for each player.
4We could, in general, also add an arbitrary 𝑏 to the function. But, we are fixing utility of disagreement at zero. Restricting attention only to utilities that assign 0 to the disagreement point fixes the location of the utilities, and we can only apply linear rather than affine transformations. But, we could have used “cardinal invariance” and allowed disagreement utility to be non-zero and the theory would be the same.
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The first three axioms are so natural that it is difficult to argue with them. Also, they do not really pin down the solution. One final axiom is used, which is key to Nash’s solution:
Axiom 4 (Independence of Irrelevant Alternatives):
For all bargaining games 𝑆 and 𝑇 with 𝑆 ⊆ 𝑇 , if 𝜑(𝑇 ) ∈ 𝑆 then 𝜑(𝑆) = 𝜑(𝑇 ).
The independence of irrelevant alternatives axiom seems to suggest some consistency about the (unmodelled) bargaining process. It is explained as follows. Suppose that the players are bargaining over a large set, and choose a particular agreement. Now suppose that some of the “disregarded” outcomes are deleted, so we have a new bargaining prob- lem with a smaller bargaining set. Given that the players disregarded these alternatives before, the independence of irrelevant alternatives axiom suggests that the solution to the new problem (with those alternatives disregarded in advance) should not change.
The axiom is very convincing. But, it does suggest that the bargaining process (which we do not know much about) is consistent. This kind of independence axiom is well-known in Microeconomics, and similar versions are used in the revealed preference approach to consumer theory. The idea follows whenever some function is being maximised (utility, social welfare, something else...). For example, if a function 𝑓 over the interval [0, 10] takes a maximum value at 5, then 𝑓 has the same maximum if we consider the smaller inteval [4, 6].
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2.4 The Nash Bargaining Solution.
We are now ready to state Nash’s solution (i.e. the solution concept proposed by Nash) to the bargaining problem:
Definition (The Nash Bargaining Solution):
The Nash bargaining solution is a solution concept that for each bargaining game 𝑆 selects the unique alternative (𝑆) = (𝑎∗1, 𝑎∗2) that solves the problem:
max 𝑎1𝑎2 subject to (𝑎1, 𝑎2) ∈ 𝑆 and 𝑎1, 𝑎2 ⩾ 0.
The product of utilities 𝑎1𝑎2 is sometimes called the “Nash product”, so Nash’s solution maximises the Nash product. Another way of writing the condition “ (𝑆) is the alterna- tive that solves the following problem” is to write:
(𝑆) = arg max𝑎∈𝑆,𝑎⩾0𝑎1𝑎2.
A very nice thing about the Nash solution is that it satisfies all of our axioms:
Proposition 1:
The Nash bargaining solution satisfies axioms 1-4 (Pareto efficiency, Symmetry, inear Invariance, and Independence of Irrelevant Alternatives).
Proof - see exercise class 8. An incredible fact is that the Nash bargaining solution is the only solution concept that satisfies axioms 1-4:
Proposition 2:
If a solution concept 𝜑 satisfies axioms 1-4 (Pareto efficiency, Symmetry, inear In- variance, and Independence of Irrelevant Alternatives), then it is the Nash bargaining solution. That is, 𝜑 = .
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Proof. Suppose that a solution concept 𝜑 satisfies axioms 1-4. Consider a bargaining prob- lem 𝑆. Apply a linear transformation to rescale the bargaining problem to 𝑏𝑆 so that the al- ternative chosen by Nash solution is mapped to (1, 1). We want to show that 𝜑(𝑏𝑆) = (1, 1):
0
S
Rescale the problem so that
!(bS ) = (1,1) 0
bS
1
1
a1a2 = 1
Utility for player 2
Utility for player 1
T
Having rescaled the problem, we now show that every point 𝑎 in the bargaining set satisfies 𝑎1 + 𝑎2 ⩽ 2. To see this, we use the fact that 𝑏𝑆 is convex. For any 𝑎 ∈ 𝑏𝑆, we know that the line connecting 𝑎 and (1, 1) is entirely contained in 𝑏𝑆:
𝛽𝑎 + (1 − 𝛽)(1, 1) = (1 + 𝛽(𝑎1 − 1), 1 + 𝛽(𝑎2 − 1)) ∈ 𝑏𝑆 for all 𝛽 ∈ [0, 1].
We know the Nash product is maximised at (1, 1), so we know that:
! 1 + 𝛽(𝑎1 − 1)
"! 1 + 𝛽(𝑎2 − 1)
" ⩽ 1 for all 𝛽 ∈ [0, 1].
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Equivalently, 𝑎1 + 𝑎2 ⩽ 2 − 𝛽(𝑎1 − 1)(𝑎2 − 1), for all 𝛽 ∈ [0, 1].
and letting 𝛽 → 0 gives the desired conclusion.
Now, construct a square 𝑇 in the region 𝑎1 +𝑎2 ⩽ 2 that is symmetric about the line 𝑎1 = 𝑎2 and large enough to contain 𝑏𝑆. Treating 𝑇 as a bargaining problem, the Pareto efficiency and Symmetry axioms imply that 𝜑(𝑇 ) = (1, 1).
0
bS
1
1
a1a2 = 1
Utility for player 2
Utility for player 1
T
!(T ) = !(bS ) = !(bS )
2
2
a1 + a2 = 2
By the Independence of Irrelevant Alternatives axiom, because 𝑏𝑆 ⊆ 𝑇 and 𝜑(𝑇 ) ∈ 𝑆, we know that 𝜑(𝑏𝑆) = (1, 1) = (𝑏𝑆). By the linear invariance axiom, 𝜑(𝑏𝑆) = 𝑏𝜑(𝑆) = 𝑏 (𝑆), so 𝜑(𝑆) = (𝑆). As 𝑆 was chosen arbitrarily, this establishes that 𝜑 = .
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3 Dividing a Dollar.
To give a simple application of Nash’s bargaining solution, we will now consider the prob- lem of “dividing a dollar” (or any amount, of any currency). Suppose that two players, 𝐴 and 𝐵 are bargaining over a surplus of size one. Each player 𝑖 = 𝐴, 𝐵 has strictly increas- ing and strictly concave utility 𝑢𝑖 with 𝑢𝑖(0) = 0. The set of alternatives for this problem is then:
𝑆 = {(𝑢𝐴(𝑥), 𝑢𝐵(𝑦)) ∶ 𝑥 ⩾ 0, 𝑦 ⩾ 0, 𝑥 + 𝑦 ⩽ 1},
with the disagreement point being (0, 0).
Let us study the Nash bargaining solution for this problem in two cases. In the first case, we will suppose that both players have the same utility. In the second case, we will suppose that player 𝐵 is more risk averse than player 𝐴.
Case 1: 𝑢𝐴 = 𝑢𝐵 = 𝑢.
The Nash bargaining solution is Pareto efficient, so selects an alternative that does not throw any money away. Hence, the Nash bargaining solution selects an alternative (𝑆) that solves:
max 0⩽𝑥⩽1
𝑢(𝑥)𝑢(1 − 𝑥)
The first order condition (using the product rule) for this problem is:
𝑢′(𝑥∗)𝑢(1 − 𝑥∗) − 𝑢(𝑥∗)𝑢′(1 − 𝑥∗) = 0
Hence, the Nash solution selects an alternative = (𝑥∗, 1 − 𝑥∗) such that: 𝑢′(𝑥∗) 𝑢(𝑥∗)
= 𝑢′(1 − 𝑥∗) 𝑢(1 − 𝑥∗)
,
hence, 𝑥∗ = 1 2 . The dollar is divided equally between the two players. Note that
we could have arrived at this conclusion immediately by noting that the problem is symmetric, hence the solution must be symmetric ( satisfies the symmetry axiom).
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Case 2: Player B is more risk averse than player A:
We know that risk aversion is equivalent to utility being strictly concave. To model the situation where one player is “more risk averse” than the other, we apply the same idea – player 𝐵’s utility is more concave than player 𝐴’s utility. More precise, there is a strictly concave function 𝑓 , with 𝑓(0) = 0, such that:
𝑢𝐵 = 𝑓◦𝑢𝐴,
which means 𝑢𝐵(𝑥) = 𝑓(𝑢(𝑥)) for all 𝑥.
In this case, the Nash bargaining solution selects an alternative (𝑆) that solves: max 0⩽𝑥⩽1
𝑢𝐴(𝑥)𝑢𝐵(1 − 𝑥) = 𝑢𝐴(𝑥)𝑓 ! 𝑢𝐴(1 − 𝑥)
" .
The first order condition for this problem is:
𝑢′𝐴(𝑥 ∗)𝑓
! 𝑢𝐴(1 − 𝑥
∗) " − 𝑢𝐴(𝑥
∗)𝑓 ′ ! 𝑢𝐴(1 − 𝑥
∗) " 𝑢′𝐴(1 − 𝑥
∗) = 0
So, we have: 𝑢′𝐴(𝑥
∗) 𝑢𝐴(𝑥∗)
= 𝑓 ′ ! 𝑢𝐴(1 − 𝑥∗)
" 𝑢′𝐴(1 − 𝑥
∗)
𝑓 ! 𝑢𝐴(1 − 𝑥∗)
" .
Because 𝑓 is strictly increasing, strictly concave, and 𝑓(0) = 0, we know that:
𝑓 ′(𝑦) ⩽ 𝑓(𝑦) 𝑦
for all 𝑦 ⩾ 0.
Therefore, we have: 𝑢′𝐴(𝑥
∗) 𝑢𝐴(𝑥∗)
⩽ 𝑢′𝐴(1 − 𝑥
∗) 𝑢𝐴(1 − 𝑥∗)
,
which means that 𝑥∗ ⩾ 1 2 . Hence, we get a nice result – risk aversion is detrimental
when bargaining! The less risk averse of the two players (𝐴 in this case) receives more than the more risk averse player.
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