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Games with Sequential Move and Subgame Perfection

Learning Objectives

Solving sequential games using backward induction

Incredible threat and Nash Equilibrium

Subgame Perfect (Nash) Equilibrium

Stackelberg model of oligopoly

More examples of sequential game

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Games with Sequential Move in Discrete Strategies

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Solving A Sequential Game

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Solving by Pruning (Backward Induction)

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Sequential Rationality

An optimal strategy should maximize the player’s expected payoff conditional on every information set where he has the move

A player should specify an optimal action from each of his information sets, even those that he does not believe (ex ante) will be reached in the game

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Incredible Threat and Nash Equilibrium

Two NE – (I, A) and (O, P)

Is (O, P) plausible?

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Subgame

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Subgame

All the branches of the tree that don’t rip off any information set

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Subgame Perfect Nash Equilibrium

A Subgame Perfect Nash Equilibrium (SPE) specifies a Nash Equilibrium in every subgame of the original game

Three NE – (UA, X); (DA, Y); (DB, Y)

Only one SPE – (UA, X)

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Example 2

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Two NE – (OA, O); (OB, O)

SPE – (OA, O)

Exercise 1

Find out all the pure strategy Nash Equilibria and all the Subgame Perfect Nash Equilibria

PAUSE HERE and find out the answers first

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Solving Exercise 1

		Player 2
		AC	AD	BC	BD
Player 1	UE	1, 4	1, 4	5, 2	5, 2
	UF	1, 4	1, 4	5, 2	5, 2
	DE	3, 3	6, 2	3, 3	6, 2
	DF	2, 0	6, 2	2, 0	6, 2

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NE:- (DE, AC); (DF, AD); (DF, BD)

SPE:- (DE, AC)

Strategic Thinking for Fun!!!!

A traveler gets lost on a deserted island and finds himself surrounded by a group of n > 0 cannibals.

Each cannibal wants to eat the traveler but, as each knows, there is a risk. A cannibal that attacks and eats the traveler would become tired and defenseless. After he eats, he would become an easy target for another cannibal (who would also become tired and defenseless after eating).

The cannibals are all hungry, but they cannot trust each other to cooperate. The cannibals happen to be well versed in game theory, so they will think before making a move.

Does the nearest cannibal, or any cannibal in the group, devour the lost traveler?

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Games with Sequential Move in Continuous Strategies

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Stackelberg (1934) Model of Oligopoly

Recall the Cournot Model of Oligopoly (Module 6)

Instead of choosing simultaneously, now suppose the firms choose the quantity sequentially

Firm 1 is the leader and firm 2 is the follower

Price that consumers are willing to pay depends on total number of bricks being sold:

Firms have the same marginal cost of production:

Stackelberg Model: Sequential Move Game

Firm i’s strategy:

Firm i’s payoff is its profit

For any , leader firm can predict follower firm’s behavior

it maximizes its payoff by choosing its best response

Knowing this, leader maximizes:

Hence, SPE is and

Any other Nash Equilibrium in Stackelberg Model?

Consider the following strategy:

Leader:

Follower:

Check that it is a NE, but it is not SPE because of incredible threats from the follower

Exercise 2: Double Marginalization

A tire manufacturer produces at a cost of $10 per unit which he sells to a retailer at $x and then the retailer in turn sells to consumers according to the following demand: . The retailer has no additional cost. Find out the SPE of this game. Calculate the joint profit.

Suppose instead the manufacturer could sell directly to consumers. Find out his profit maximizing q and calculate the profit.

Compare the two profits. The difference comes from double marginalization problem (having two monopolists maximizing respective profits versus one)

Solving Double Marginalization Game

For solution, please watch the video presentation