Question

3. General Extensive Form Game D Suppose the following general extensive form game 1/2 1/2 (2, 2) (2, 2) (0, 6) (6, 0 (0,0 (6, 4) (a) Represent this game in normal form by using a matrix, and find all pure strategy Bayesian Nash equilibrium (equilibria) b) Find pure strategy subgame perfect equilibrium (or equilibria) of this game. c) Find pure strategy perfect Bayesian equilibrium (or equilibria) of this game.

Answer 1

USING THE BAYESIAN NASH EQUILIBRIUM EXTENSIVE FORM GAME:

An extensive-form game can contain a part that could be considered a smaller game in itself; such a smaller game that is embedded in a larger game is called a subgame. A main property of backward induction is that, when restricted to a subgame of the game, the equilibrium computed using backward induction remains an equilibrium (computed again via backward induction) of the subgame. Subgame perfection generalizes this notion to general dynamic games. A Nash equilibrium is said to be subgame perfect if and only if it is a Nash equilibrium in every subgame of the game. A subgame must be a well-defined game when it is considered separately i.e.
• it must contain an initial node.
• all the moves and information sets from that node must remain in the subgame.

Note that, in each subgame, the equilibrium computed via backward induction remains to be an equilibrium of the subgame. Any subgame other than the entire game itself is called proper.

(a) Using the normal form, finding all pure-strategies of Bayesian Nash equilibrum in matrix form. A strategy is in Nash equilibrum if no single player can gain by deviating from the strategy. In extensive form games, the notion of Nash equilibrum remains the same if no single player can gain by deviating in any way from the actions prescribed to him by the strategy then the strategy is in Nash equilibrum. The problem in extensive form games is that players act in turns. This introduces what we call "empty threats". In some cases, a player can deviate from his strategy because he knows the other player will see this and be forced to deviate as well resulting in an overall profit for the first player. Hence the notion of a single player deviation from the overall strategy is too weak for extensive form games. SPE will eliminate these empty threats by requiring ideal play in any sub-game. The players find themselves in, i.e. a strategy which is a "best response"
at any history.

There are two ways of finding a pure-strategy Bayesian Nash Equilibrium (BNE):
Method 1: This method works directly on the normal form representation, which is most easily done by converting the game into the corresponding payoff matrix. This is by simply finding the Nash equilibria from the payoff matrix. This method computes expected payoffs from an ex-ante perspective before the players learns their types. Notice that, if the set of actions available or the set of possible types is infinite, we cannot construct the payoff matrix so method 1 will not work so the payoff matrix for the Entry Game with Cost Uncertainty with best responses will be marked with a star.

Method 2. This method for finding the BNE converts the game into an equivalent "bigger" game in which the different types of each player are treated as separate players. The payoff to player (1, 2) is the expected payoff of player 1, conditional on being type 2. Any NE of the bigger game is a BNE of the original game and vice versa. The payoff in the Bayesian normal-form matrix is the summation over all types of the probability of a type multiplied by the expected payoff conditional on that type. If player 1 is best responding in the original game then it is impossible to increase his payoff conditional on any of his types (or else this summation would be higher) so each player (1, 2) must be best responding in the bigger game.

Here we have 3 Nash equilibria (recall that, in extensive form games a strategy has to be specified for every node in the game tree, even if the strategy does not reach it).

1. NE1: S1(∈) = L, S2(h) = R0 ∀h ∈ {H, L}.

2. NE2: S1(∈) = R, S2(h) = L 0 ∀h ∈ {AB, CD}

3. NE3: S1(∈) = R, S2(h) = R 0 ∀h ∈ {XY, XY}

LEFT SIDE

	L	CD	XY
H	(1, (1/2), (1, (1/2)	2, (2,2) (1, (1/2)	(0,0) (6,4)
AB	(1, (1/2), (2, (2,2)	2, (2,2) 2, (2,2)	(0,0) (6,4)
XY	(0,6) (6,0)	(0,0) (6,4)	( ) ( )

Let’s look at NE1 above: If player 1 deviates and goes right, the strategy he dictates he will lose more as it reaches as a result of (0,0). However in extensive form games this is an empty threat because player 2 will see this and go left, resulting in overall profit for player 2, so rationally it is in fact profitable for player 1 to deviate from his strategy in this case.

(b) The pure-strategy subgame equilibria of this game is by backwards induction, we can see that the only subgame Nash equilibria is for player 1 to play (End, End) and for player 2 to also play (End, End). An extensive-form game can contain a part that could be considered a smaller game in itself such a smaller game that is embedded in a larger game is called a subgame. A main property of backward induction is that when restricted to a subgame of the game, the equilibrium computed using backward induction remains an equilibrium (computed again via backward induction) of the subgame. Subgame generalizes this notion to general dynamic games. Using backwards induction, we know that player 2 will play A if player 1 plays either Give or Share, and be indifferent between A and B if player 1 plays Keep hence there are 2 sub-game equilibria.

(c) The pure-strategy perfect Bayesian equilibria of this game. Perfect Bayesian equilibrium (PBE) was invented in order to refine Bayesian Nash equilibrium in a way that is similar to how subgame-perfect Nash equilibrium refines Nash equilibrium. Consider the following game of complete, but imperfect information. First, player 1 chooses among three actions: H,A, and X. If player 1 chooses A then the game ends without a move by player 2. If player 1 chooses either B or Y then player 2 learns that A was not chosen ( but not which of X or Y was chosen) and then chooses between two actions A' and B', after which the game ends. Payoffs are given in the extensive form.

Using the normal form representation of this game given below we see that there are two pure strategy Nash-equilibria - (L,L') and (R,R'). To determine which of these Nash equilibria are subgame perfect, we use the extensive form representation to define the game's subgames. So the game above has no proper subgames and the requirement of subgame perfection is trivially satisfied, and is just the Nash equilibrium of the whole game. So in the game above both (L,L') and (R,R') are subgame perfect Nash equilibria. However, one can see that (R,R') clearly depends on a noncredible threat: if player 2 gets the move, then playing L' dominates playing R', so player 1 should not be induced to play R by 2's threat to play R' given the move.

To strengthen the equilibrium concept to rule out the subgame perfect Nash equilibrium (R,R') we impose the following requirements.

R1: At each information set, the player with the move must have a belief about which node in the information set has been reached by the play of the game. For a nonsingleton information set, a belief is a probability distribution over the nodes in the information set; for a singleton information set, the player's belief puts one on the decision node.

R2: Given the beliefs, the players' strategies must be sequentially rational. That is at each information set the action taken by the player with the move (and the player's subsequent strategy) must be optimal given the player's belief at the information set and the other players' subsequent strategies ( where a "subsequent strategy" is a complete plan of action covering every contingency that might arise after the given information set has been reached).

R3: At information sets on the equilibrium path, beliefs are determined by Bayes' rule and the players' equilibrium strategies.

R4: At information sets off the equilibrium path, beliefs are determined by Bayes' rule and the players' equilibrium strategies where possible.