Game Applications Chapter 29

Nash Equilibrium In any Nash equilibrium (NE) each player chooses a “best” response to the choices made by all of the other players. A game may have more than one NE. How can we locate every one of a game’s Nash equilibria? If there is more than one NE, can we argue that one is more likely to occur than another?

Best Responses Think of a 2×2 game; i.e., a game with two players, A and B, each with two actions. A can choose between actions a A 1 and a A 2. B can choose between actions a B 1 and a B 2. There are 4 possible action pairs; (a A 1, a B 1 ), (a A 1, a B 2 ), (a A 2, a B 1 ), (a A 2, a B 2 ). Each action pair will usually cause different payoffs for the players.

Best Responses Suppose that A’s and B’s payoffs when the chosen actions are a A 1 and a B 1 are U A (a A 1, a B 1 ) = 6 and U B (a A 1, a B 1 ) = 4. Similarly, suppose that U A (a A 1, a B 2 ) = 3 and U B (a A 1, a B 2 ) = 5 U A (a A 2, a B 1 ) = 4 and U B (a A 2, a B 1 ) = 3 U A (a A 2, a B 2 ) = 5 and U B (a A 2, a B 2 ) = 7.

Best Responses U A (a A 1, a B 1 ) = 6 and U B (a A 1, a B 1 ) = 4 U A (a A 1, a B 2 ) = 3 and U B (a A 1, a B 2 ) = 5 U A (a A 2, a B 1 ) = 4 and U B (a A 2, a B 1 ) = 3 U A (a A 2, a B 2 ) = 5 and U B (a A 2, a B 2 ) = 7.

Best Responses U A (a A 1, a B 1 ) = 6 and U B (a A 1, a B 1 ) = 4 U A (a A 1, a B 2 ) = 3 and U B (a A 1, a B 2 ) = 5 U A (a A 2, a B 1 ) = 4 and U B (a A 2, a B 1 ) = 3 U A (a A 2, a B 2 ) = 5 and U B (a A 2, a B 2 ) = 7. If B chooses action a B 1 then A’s best response is ??

Best Responses U A (a A 1, a B 1 ) = 6 and U B (a A 1, a B 1 ) = 4 U A (a A 1, a B 2 ) = 3 and U B (a A 1, a B 2 ) = 5 U A (a A 2, a B 1 ) = 4 and U B (a A 2, a B 1 ) = 3 U A (a A 2, a B 2 ) = 5 and U B (a A 2, a B 2 ) = 7. If B chooses action a B 1 then A’s best response is action a A 1 (because 6 > 4).

Best Responses U A (a A 1, a B 1 ) = 6 and U B (a A 1, a B 1 ) = 4 U A (a A 1, a B 2 ) = 3 and U B (a A 1, a B 2 ) = 5 U A (a A 2, a B 1 ) = 4 and U B (a A 2, a B 1 ) = 3 U A (a A 2, a B 2 ) = 5 and U B (a A 2, a B 2 ) = 7. If B chooses action a B 1 then A’s best response is action a A 1 (because 6 > 4). If B chooses action a B 2 then A’s best response is ??

Best Responses U A (a A 1, a B 1 ) = 6 and U B (a A 1, a B 1 ) = 4 U A (a A 1, a B 2 ) = 3 and U B (a A 1, a B 2 ) = 5 U A (a A 2, a B 1 ) = 4 and U B (a A 2, a B 1 ) = 3 U A (a A 2, a B 2 ) = 5 and U B (a A 2, a B 2 ) = 7. If B chooses action a B 1 then A’s best response is action a A 1 (because 6 > 4). If B chooses action a B 2 then A’s best response is action a A 2 (because 5 > 3).

Best Responses If B chooses a B 1 then A chooses a A 1. If B chooses a B 2 then A chooses a A 2. A’s best-response “curve” is therefore A’s best response aA1aA1 aA2aA2 aB2aB2 aB1aB1 B’s action + +

Best Responses U A (a A 1, a B 1 ) = 6 and U B (a A 1, a B 1 ) = 4 U A (a A 1, a B 2 ) = 3 and U B (a A 1, a B 2 ) = 5 U A (a A 2, a B 1 ) = 4 and U B (a A 2, a B 1 ) = 3 U A (a A 2, a B 2 ) = 5 and U B (a A 2, a B 2 ) = 7.

Best Responses U A (a A 1, a B 1 ) = 6 and U B (a A 1, a B 1 ) = 4 U A (a A 1, a B 2 ) = 3 and U B (a A 1, a B 2 ) = 5 U A (a A 2, a B 1 ) = 4 and U B (a A 2, a B 1 ) = 3 U A (a A 2, a B 2 ) = 5 and U B (a A 2, a B 2 ) = 7. If A chooses action a A 1 then B’s best response is ??

Best Responses U A (a A 1, a B 1 ) = 6 and U B (a A 1, a B 1 ) = 4 U A (a A 1, a B 2 ) = 3 and U B (a A 1, a B 2 ) = 5 U A (a A 2, a B 1 ) = 4 and U B (a A 2, a B 1 ) = 3 U A (a A 2, a B 2 ) = 5 and U B (a A 2, a B 2 ) = 7. If A chooses action a A 1 then B’s best response is action a B 2 (because 5 > 4).

Best Responses U A (a A 1, a B 1 ) = 6 and U B (a A 1, a B 1 ) = 4 U A (a A 1, a B 2 ) = 3 and U B (a A 1, a B 2 ) = 5 U A (a A 2, a B 1 ) = 4 and U B (a A 2, a B 1 ) = 3 U A (a A 2, a B 2 ) = 5 and U B (a A 2, a B 2 ) = 7. If A chooses action a A 1 then B’s best response is action a B 2 (because 5 > 4). If A chooses action a A 2 then B’s best response is ??.

Best Responses U A (a A 1, a B 1 ) = 6 and U B (a A 1, a B 1 ) = 4 U A (a A 1, a B 2 ) = 3 and U B (a A 1, a B 2 ) = 5 U A (a A 2, a B 1 ) = 4 and U B (a A 2, a B 1 ) = 3 U A (a A 2, a B 2 ) = 5 and U B (a A 2, a B 2 ) = 7. If A chooses action a A 1 then B’s best response is action a B 2 (because 5 > 4). If A chooses action a A 2 then B’s best response is action a B 2 (because 7 > 3).

Best Responses If A chooses a A 1 then B chooses a B 2. If A chooses a A 2 then B chooses a B 2. B’s best-response “curve” is therefore A’s action aA1aA1 aA2aA2 aB2aB2 aB1aB1 B’s best response

Best Responses If A chooses a A 1 then B chooses a B 2. If A chooses a A 2 then B chooses a B 2. B’s best-response “curve” is therefore A’s action aA1aA1 aA2aA2 aB2aB2 aB1aB1 B’s best response Notice that a B 2 is a strictly dominant action for B.

Best Responses & Nash Equilibria A’s response aA1aA1 aA2aA2 aB2aB2 aB1aB1 aA1aA1 aA2aA2 aB2aB2 aB1aB1 + + A’s choice B’s choiceB’s response How can the players’ best-response curves be used to locate the game’s Nash equilibria? B A

Best Responses & Nash Equilibria A’s response aA1aA1 aA2aA2 aB2aB2 aB1aB1 aA1aA1 aA2aA2 aB2aB2 aB1aB1 + + A’s choice B’s choiceB’s response How can the players’ best-response curves be used to locate the game’s Nash equilibria? Put one curve on top of the other. B A

Best Responses & Nash Equilibria A’s response aA1aA1 aA2aA2 aB2aB2 aB1aB1 aA1aA1 aA2aA2 aB2aB2 aB1aB1 + + A’s choice B’s choiceB’s response B A

How can the players’ best-response curves be used to locate the game’s Nash equilibria? Put one curve on top of the other. Best Responses & Nash Equilibria A’s response aA1aA1 aA2aA2 aB2aB2 aB1aB1 + + B’s response Is there a Nash equilibrium?

How can the players’ best-response curves be used to locate the game’s Nash equilibria? Put one curve on top of the other. Best Responses & Nash Equilibria A’s response aA1aA1 aA2aA2 aB2aB2 aB1aB1 + + Is there a Nash equilibrium? Yes, (a A 2, a B 2 ). Why? B’s response

How can the players’ best-response curves be used to locate the game’s Nash equilibria? Put one curve on top of the other. Best Responses & Nash Equilibria A’s response aA1aA1 aA2aA2 aB2aB2 aB1aB1 + + Is there a Nash equilibrium? Yes, (a A 2, a B 2 ). Why? a A 2 is a best response to a B 2. a B 2 is a best response to a A 2. B’s response

Best Responses & Nash Equilibria 6,43,5 5,75,74,3 aA1aA1 aA2aA2 aB1aB1 aB2aB2 Player B Player A a A 2 is the only best response to a B 2. a B 2 is the only best response to a A 2. Here is the strategic form of the game.

Best Responses & Nash Equilibria 6,43,5 5,75,74,3 aA1aA1 aA2aA2 aB1aB1 aB2aB2 Player B Player A Here is the strategic form of the game. a A 2 is the only best response to a B 2. a B 2 is the only best response to a A 2. Is there a 2 nd Nash eqm.?

Best Responses & Nash Equilibria 6,43,5 5,75,74,3 aA1aA1 aA2aA2 aB1aB1 aB2aB2 Player B Player A Is there a 2 nd Nash eqm.? No, because a B 2 is a strictly dominant action for Player B. a A 2 is the only best response to a B 2. a B 2 is the only best response to a A 2. Here is the strategic form of the game.

Best Responses & Nash Equilibria Now allow both players to randomize (i.e., mix) over their actions. 6,43,5 5,74,3 aA1aA1 aA2aA2 aB1aB1 aB2aB2 Player B Player A

Best Responses & Nash Equilibria Now allow both players to randomize (i.e., mix) over their actions. 6,43,5 5,74,3 aA1aA1 aA2aA2 aB1aB1 aB2aB2 Player B Player A  A 1 is the prob. A chooses action a A 1.  B 1 is the prob. B chooses action a B 1.

Best Responses & Nash Equilibria 6,43,5 5,74,3 aA1aA1 aA2aA2 aB1aB1 aB2aB2 Player B Player A  A 1 is the prob. A chooses action a A 1.  B 1 is the prob. B chooses action a B 1. Given  B 1, what value of  A 1 is best for A?

Best Responses & Nash Equilibria 6,43,5 5,74,3 aA1aA1 aA2aA2 aB1aB1 aB2aB2 Player B Player A EV A (a A 1 ) = 6  B 1 + 3(1 -  B 1 ) =  B 1.  A 1 is the prob. A chooses action a A 1.  B 1 is the prob. B chooses action a B 1. Given  B 1, what value of  A 1 is best for A?

Best Responses & Nash Equilibria 6,43,5 5,74,3 aA1aA1 aA2aA2 aB1aB1 aB2aB2 Player B Player A  A 1 is the prob. A chooses action a A 1.  B 1 is the prob. B chooses action a B 1. Given  B 1, what value of  A 1 is best for A? EV A (a A 1 ) = 6  B 1 + 3(1 -  B 1 ) =  B 1. EV A (a A 2 ) = 4  B 1 + 5(1 -  B 1 ) = 5 -  B 1.

Best Responses & Nash Equilibria  A 1 is the prob. A chooses action a A 1.  B 1 is the prob. B chooses action a B 1. Given  B 1, what value of  A 1 is best for A? EV A (a A 1 ) =  B 1. EV A (a A 2 ) = 5 -  B  B  B 1 as  B 1 ?? >=<>=< >=<>=<

Best Responses & Nash Equilibria  A 1 is the prob. A chooses action a A 1.  B 1 is the prob. B chooses action a B 1. Given  B 1, what value of  A 1 is best for A? EV A (a A 1 ) =  B 1. EV A (a A 2 ) = 5 -  B  B  B 1 as  B 1 ½. >=<>=< >=<>=<

Best Responses & Nash Equilibria  A 1 is the prob. A chooses action a A 1.  B 1 is the prob. B chooses action a B 1. Given  B 1, what value of  A 1 is best for A? EV A (a A 1 ) =  B 1. EV A (a A 2 ) = 5 -  B  B  B 1 as  B 1 ½. A’s best response is: a A 1 if  B 1 > ½ a A 2 if  B 1 < ½ a A 1 or a A 2 if  B 1 = ½ >=<>=< >=<>=<

Best Responses & Nash Equilibria  A 1 is the prob. A chooses action a A 1.  B 1 is the prob. B chooses action a B 1. Given  B 1, what value of  A 1 is best for A? EV A (a A 1 ) =  B 1. EV A (a A 2 ) = 5 -  B  B  B 1 as  B 1 ½. A’s best response is: a A 1 (i.e.  A 1 = 1) if  B 1 > ½ a A 2 (i.e.  A 1 = 0) if  B 1 < ½ a A 1 or a A 2 (i.e. 0   A 1  1) if  B 1 = ½ >=<>=< >=<>=<

Best Responses & Nash Equilibria 0 A1A1 1 B1B1 0 A’s best response ½ A’s best response is: a A 1 (i.e.  A 1 = 1) if  B 1 > ½ a A 2 (i.e.  A 1 = 0) if  B 1 < ½ a A 1 or a A 2 (i.e. 0   A 1  1) if  B 1 = ½ 1

Best Responses & Nash Equilibria 0 A1A1 1 B1B1 0 A’s best response ½ 1 A’s best response is: a A 1 (i.e.  A 1 = 1) if  B 1 > ½ a A 2 (i.e.  A 1 = 0) if  B 1 < ½ a A 1 or a A 2 (i.e. 0   A 1  1) if  B 1 = ½

Best Responses & Nash Equilibria 0 A1A1 1 B1B1 0 A’s best response ½ 1 A’s best response is: a A 1 (i.e.  A 1 = 1) if  B 1 > ½ a A 2 (i.e.  A 1 = 0) if  B 1 < ½ a A 1 or a A 2 (i.e. 0   A 1  1) if  B 1 = ½

Best Responses & Nash Equilibria 0 A1A1 1 B1B1 0 A’s best response ½ 1 This is A’s best response curve when players are allowed to mix over their actions. A’s best response is: a A 1 (i.e.  A 1 = 1) if  B 1 > ½ a A 2 (i.e.  A 1 = 0) if  B 1 < ½ a A 1 or a A 2 (i.e. 0   A 1  1) if  B 1 = ½

Best Responses & Nash Equilibria 6,43,5 5,74,3 aA1aA1 aA2aA2 aB1aB1 aB2aB2 Player B Player A  A 1 is the prob. A chooses action a A 1.  B 1 is the prob. B chooses action a B 1. Given  A 1, what value of  B 1 is best for B?

Best Responses & Nash Equilibria 6,43,5 5,74,3 aA1aA1 aA2aA2 aB1aB1 aB2aB2 Player B Player A  A 1 is the prob. A chooses action a A 1.  B 1 is the prob. B chooses action a B 1. Given  A 1, what value of  B 1 is best for B? EV B (a B 1 ) = 4  A 1 + 3(1 -  A 1 ) = 3 +  A 1.

Best Responses & Nash Equilibria 6,43,5 5,74,3 aA1aA1 aA2aA2 aB1aB1 aB2aB2 Player B Player A  A 1 is the prob. A chooses action a A 1.  B 1 is the prob. B chooses action a B 1. Given  A 1, what value of  B 1 is best for B? EV B (a B 1 ) = 4  A 1 + 3(1 -  A 1 ) = 3 +  A 1. EV B (a B 2 ) = 5  A 1 + 7(1 -  A 1 ) =  A 1.

Best Responses & Nash Equilibria  A 1 is the prob. A chooses action a A 1.  B 1 is the prob. B chooses action a B 1. Given  A 1, what value of  B 1 is best for B? EV B (a B 1 ) = 3 +  A 1. EV B (a B 2 ) =  A  A  A 1 as  A 1 ?? >=<>=< >=<>=<

Best Responses & Nash Equilibria  A 1 is the prob. A chooses action a A 1.  B 1 is the prob. B chooses action a B 1. Given  A 1, what value of  B 1 is best for B? EV B (a B 1 ) = 3 +  A 1. EV B (a B 2 ) =  A  A 1 <  A 1 for all 0   A 1  1.

Best Responses & Nash Equilibria  A 1 is the prob. A chooses action a A 1.  B 1 is the prob. B chooses action a B 1. Given  B 1, what value of  A 1 is best for A? EV B (a B 1 ) = 3 +  A 1. EV B (a B 2 ) =  A  A 1 <  A 1 for all 0   A 1  1. B’s best response is: a B 2 always (i.e.  B 1 = 0 always).

Best Responses & Nash Equilibria 0 A1A1 1 B1B1 0 B’s best response ½ B’s best response is a B 2 always (i.e.  B 1 = 0 always). 1 This is B’s best response curve when players are allowed to mix over their actions.

Best Responses & Nash Equilibria 0 A1A1 1 B1B1 0 B’s best response ½ 1 0 A1A1 1 B1B1 0 A’s best response ½ 1 BA

Best Responses & Nash Equilibria 0 A1A1 1 B1B1 0 B’s best response ½ 1 0 A1A1 1 B1B1 0 A’s best response ½ 1 BA Is there a Nash equilibrium?

Best Responses & Nash Equilibria 0 A1A1 1 B1B1 0 B’s best response ½ 1 0 A1A1 1 B1B1 0 A’s best response ½ 1 BA Is there a Nash equilibrium?

Best Responses & Nash Equilibria 0 A1A1 1 B1B1 0 B’s best response ½ 1 A’s best response Is there a Nash equilibrium?

Best Responses & Nash Equilibria 0 A1A1 1 B1B1 0 B’s best response ½ 1 A’s best response Is there a Nash equilibrium? Yes. Just one. (  A 1,  B 1 ) = (0,0); i.e. A chooses a A 2 only & B chooses a B 2 only.

Best Responses & Nash Equilibria 6,43,5 5,74,3 aA1aA1 aA2aA2 aB1aB1 aB2aB2 Player B Player A Let’s change the game.

3,1 Best Responses & Nash Equilibria 6,43,5 5,74,3 aA1aA1 aA2aA2 aB1aB1 aB2aB2 Player B Player A Here is a new 2×2 game.

3,1 Best Responses & Nash Equilibria 6,4 5,74,3 aA1aA1 aA2aA2 aB1aB1 aB2aB2 Player B Player A Here is a new 2×2 game. Again let  A 1 be the prob. that A chooses a A 1 and let  B 1 be the prob. that B chooses a B 1. What are the NE of this game? Notice that Player B no longer has a strictly dominant action.

3,1 Best Responses & Nash Equilibria 6,4 5,74,3 aA1aA1 aA2aA2 aB1aB1 aB2aB2 Player B Player A  A 1 is the prob. that A chooses a A 1.  B 1 is the prob. that B chooses a B 1. EV A (a A 1 ) = ?? EV A (a A 2 ) = ??

3,1 Best Responses & Nash Equilibria 6,4 5,74,3 aA1aA1 aA2aA2 aB1aB1 aB2aB2 Player B Player A  A 1 is the prob. that A chooses a A 1.  B 1 is the prob. that B chooses a B 1. EV A (a A 1 ) = 6  B 1 + 3(1 -  B 1 ) =  B 1. EV A (a A 2 ) = ??

3,1 Best Responses & Nash Equilibria 6,4 5,74,3 aA1aA1 aA2aA2 aB1aB1 aB2aB2 Player B Player A  A 1 is the prob. that A chooses a A 1.  B 1 is the prob. that B chooses a B 1. EV A (a A 1 ) = 6  B 1 + 3(1 -  B 1 ) =  B 1. EV A (a A 2 ) = 4  B 1 + 5(1 -  B 1 ) = 5 -  B 1.

3,1 Best Responses & Nash Equilibria 6,4 5,74,3 aA1aA1 aA2aA2 aB1aB1 aB2aB2 Player B Player A  A 1 is the prob. that A chooses a A 1.  B 1 is the prob. that B chooses a B 1. EV A (a A 1 ) = 6  B 1 + 3(1 -  B 1 ) =  B 1. EV A (a A 2 ) = 4  B 1 + 5(1 -  B 1 ) = 5 -  B  B  B 1 as  B 1 ½. >=<>=< >=<>=<

Best Responses & Nash Equilibria EV A (a A 1 ) = 6  B 1 + 3(1 -  B 1 ) =  B 1. EV A (a A 2 ) = 4  B 1 + 5(1 -  B 1 ) = 5 -  B  B  B 1 as  B 1 ½. >=<>=< >=<>=< 0 A1A1 1 B1B1 0 A’s best response ½ 1

Best Responses & Nash Equilibria EV A (a A 1 ) = 6  B 1 + 3(1 -  B 1 ) =  B 1. EV A (a A 2 ) = 4  B 1 + 5(1 -  B 1 ) = 5 -  B  B  B 1 as  B 1 ½. >=<>=< >=<>=< 0 A1A1 1 B1B1 0 A’s best response ½ 1

3,1 Best Responses & Nash Equilibria 6,4 5,74,3 aA1aA1 aA2aA2 aB1aB1 aB2aB2 Player B Player A  A 1 is the prob. that A chooses a A 1.  B 1 is the prob. that B chooses a B 1. EV B (a B 1 ) = ?? EV B (a B 2 ) = ??

3,1 Best Responses & Nash Equilibria 6,4 5,74,3 aA1aA1 aA2aA2 aB1aB1 aB2aB2 Player B Player A  A 1 is the prob. that A chooses a A 1.  B 1 is the prob. that B chooses a B 1. EV B (a B 1 ) = 4  A 1 + 3(1 -  A 1 ) = 3 +  A 1. EV B (a B 2 ) = ??

3,1 Best Responses & Nash Equilibria 6,4 5,74,3 aA1aA1 aA2aA2 aB1aB1 aB2aB2 Player B Player A  A 1 is the prob. that A chooses a A 1.  B 1 is the prob. that B chooses a B 1. EV B (a B 1 ) = 4  A 1 + 3(1 -  A 1 ) = 4 +  A 1. EV B (a B 2 ) =  A 1 + 7(1 -  A 1 ) =  A 1.

3,1 Best Responses & Nash Equilibria 6,4 5,74,3 aA1aA1 aA2aA2 aB1aB1 aB2aB2 Player B Player A  A 1 is the prob. that A chooses a A 1.  B 1 is the prob. that B chooses a B 1. EV B (a B 1 ) = 4  A 1 + 3(1 -  A 1 ) = 3 +  A 1. EV B (a B 2 ) =  A 1 + 7(1 -  A 1 ) =  A  A  A 1 as  A 1. >=<>=< >=<>=< 4 7 /

Best Responses & Nash Equilibria EV B (a B 1 ) = 4  A 1 + 3(1 -  A 1 ) = 3 +  A 1. EV B (a B 2 ) =  A 1 + 7(1 -  A 1 ) =  A  A  A 1 as  A 1. >=<>=< >=<>=< 4 7 / 0 A1A1 1 B1B / B’s best response

Best Responses & Nash Equilibria EV B (a B 1 ) = 4  A 1 + 3(1 -  A 1 ) = 3 +  A 1. EV B (a B 2 ) =  A 1 + 7(1 -  A 1 ) =  A  A  A 1 as  A 1. >=<>=< >=<>=< 4 7 / 0 A1A1 1 B1B / B’s best response

Best Responses & Nash Equilibria 0 A1A1 1 B1B1 0 B’s best response 1 0 A1A1 1 B1B1 0 A’s best response ½ 1 BA 4 7 /

Best Responses & Nash Equilibria 0 A1A1 1 B1B1 0 B’s best response 1 0 A1A1 1 B1B1 0 A’s best response ½ 1 BA 4 7 / Is there a Nash equilibrium?

Best Responses & Nash Equilibria 0 A1A1 1 B1B1 0 B’s best response 1 BA 4 7 / 0 A1A1 1 B1B1 0 A’s best response ½ 1 Is there a Nash equilibrium?

Best Responses & Nash Equilibria 0 A1A1 1 B1B1 0 B’s best response / A’s best response ½ Is there a Nash equilibrium?

Best Responses & Nash Equilibria 0 A1A1 1 B1B1 0 B’s best response / A’s best response ½ Is there a Nash equilibrium? Yes. 3 of them.

Best Responses & Nash Equilibria 0 A1A1 1 B1B1 0 B’s best response / A’s best response ½ Is there a Nash equilibrium? Yes. 3 of them. (  A 1,  B 1 ) = (0,0)

Best Responses & Nash Equilibria 0 A1A1 1 B1B1 0 B’s best response / A’s best response ½ Is there a Nash equilibrium? Yes. 3 of them. (  A 1,  B 1 ) = (0,0) (  A 1,  B 1 ) = (1,1) Is there a Nash equilibrium?

Best Responses & Nash Equilibria 0 A1A1 1 B1B1 0 B’s best response / A’s best response ½ Is there a Nash equilibrium? Yes. 3 of them. (  A 1,  B 1 ) = (0,0) (  A 1,  B 1 ) = (1,1) (  A 1,  B 1 ) = (, ) ½ 4 7 / Is there a Nash equilibrium?

Some Important Types of Games Games of coordination Games of competition Games of coexistence Games of commitment Bargaining games

Coordination Games Simultaneous play games in which the payoffs to the players are largest when they coordinate their actions. Famous examples are: The Battle of the Sexes Game The Prisoner’s Dilemma Game Assurance Games Chicken

Coordination Games; The Battle of the Sexes Sissy prefers watching ballet to watching mud wrestling. Jock prefers watching mud wrestling to watching ballet. Both prefer watching something together to being apart.

Coordination Games; The Battle of the Sexes 1,28,4 4,82,1 B MW B Jock Sissy  S B is the prob. that Sissy chooses ballet.  J B is the prob. that Jock chooses ballet.

Coordination Games; The Battle of the Sexes 1,28,4 4,82,1 B MW B Jock Sissy  S B is the prob. that Sissy chooses ballet.  J B is the prob. that Jock chooses ballet. What are the players’ best-response functions?

Coordination Games; The Battle of the Sexes 1,28,4 4,82,1 B MW B Jock Sissy  S B is the prob. that Sissy chooses ballet.  J B is the prob. that Jock chooses ballet. What are the players’ best-response functions? EV S (B) = 8  J B + (1 -  J B ) =  J B.

Coordination Games; The Battle of the Sexes 1,28,4 4,82,1 B MW B Jock Sissy  S B is the prob. that Sissy chooses ballet.  J B is the prob. that Jock chooses ballet. What are the players’ best-response functions? EV S (B) = 8  J B + (1 -  J B ) =  J B. EV S (MW) = 2  J B + 4(1 -  J B ) =  J B.

EV S (B) = 8  J B + (1 -  J B ) =  J B. EV S (MW) = 2  J B + 4(1 -  J B ) =  J B  J B  J B as  J B. 1 3 / Coordination Games; The Battle of the Sexes 1,28,4 4,82,1 B MW B Jock Sissy  S B is the prob. that Sissy chooses ballet.  J B is the prob. that Jock chooses ballet. What are the players’ best-response functions? >=<>=< >=<>=<

EV S (B) = 8  J B + (1 -  J B ) =  J B. EV S (MW) = 2  J B + 4(1 -  J B ) =  J B  J B  J B as  J B. 1 3 / Coordination Games; The Battle of the Sexes 1,28,4 4,82,1 B MW B Jock Sissy  S B is the prob. that Sissy chooses ballet.  J B is the prob. that Jock chooses ballet. >=<>=< >=<>=< SBSB JBJB /

EV S (B) = 8  J B + (1 -  J B ) =  J B. EV S (MW) = 2  J B + 4(1 -  J B ) =  J B  J B  J B as  J B. 1 3 / Coordination Games; The Battle of the Sexes 1,28,4 4,82,1 B MW B Jock Sissy  S B is the prob. that Sissy chooses ballet.  J B is the prob. that Jock chooses ballet. >=<>=< >=<>=< SBSB JBJB / Sissy

Coordination Games; The Battle of the Sexes SBSB JBJB / Sissy SBSB JBJB / Jock

Coordination Games; The Battle of the Sexes SBSB JBJB / Sissy SBSB JBJB / Jock The game’s NE are ??

Coordination Games; The Battle of the Sexes SBSB JBJB / Sissy SBSB JBJB / Jock The game’s NE are ??

Coordination Games; The Battle of the Sexes SBSB JBJB / Sissy Jock The game’s NE are ?? 1 3 /

Coordination Games; The Battle of the Sexes SBSB JBJB / Sissy Jock The game’s NE are: (  J B,  S B ) = (0, 0); i.e., (MW, MW) 1 3 /

Coordination Games; The Battle of the Sexes SBSB JBJB / Sissy Jock The game’s NE are: (  J B,  S B ) = (0, 0); i.e., (MW, MW) (  J B,  S B ) = (1, 1); i.e., (B, B) 1 3 /

Coordination Games; The Battle of the Sexes SBSB JBJB / Sissy Jock The game’s NE are: (  J B,  S B ) = (0, 0); i.e., (MW, MW) (  J B,  S B ) = (1, 1); i.e., (B, B) (  J B,  S B ) = (, ); i.e., both watch the ballet with prob. 1/9, both watch the mud wrestling with prob. 4/9, and with prob. 4/9 they watch different events. 1 3 / 1 3 / 1 3 /

Coordination Games; The Battle of the Sexes 1,28,4 4,82,1 B MW B Jock Sissy  S B is the prob. that Sissy chooses ballet.  J B is the prob. that Jock chooses ballet. For Sissy the expected value of the NE (  J B,  S B ) = (, ) is 8× + 1× + 2× + 4× = < 4 and / 2 9 / 2 9 / 4 9 / 10 3 / 1 3 / 1 3 /

Coordination Games; The Battle of the Sexes 1,28,4 4,82,1 B MW B Jock Sissy  S B is the prob. that Sissy chooses ballet.  J B is the prob. that Jock chooses ballet. For Sissy the expected value of the NE (  J B,  S B ) = (, ) is 8× + 1× + 2× + 4× = < 4 and / 2 9 / 2 9 / 4 9 / 10 3 / 1 3 / 1 3 / For Jock the expected value of the NE (  J B,  S B ) = (, ) is 4× + 2× + 1× + 8× = ; 4 < < / 2 9 / 2 9 / 4 9 / 14 3 / 1 3 / 1 3 / / 3

Coordination Games; The Battle of the Sexes 1,28,4 4,82,1 B MW B Jock Sissy  S B is the prob. that Sissy chooses ballet.  J B is the prob. that Jock chooses ballet. For Sissy the expected value of the NE (  J B,  S B ) = (, ) is 8× + 1× + 2× + 4× = < 4 and / 2 9 / 2 9 / 4 9 / 10 3 / 1 3 / 1 3 / For Jock the expected value of the NE (  J B,  S B ) = (, ) is 4× + 2× + 1× + 8× = ; 4 < < / 2 9 / 2 9 / 4 9 / 14 3 / 1 3 / 1 3 / / 3 So, is the mixed strategy NE a focal point for the game?

Coordination Games; The Prisoner’s Dilemma A simultaneous play game in which each player has a strictly dominant action. The only NE, therefore, is the choice by each player of her strictly dominant action. Yet both players can achieve strictly larger payoffs than in the NE by coordinating with each other on another pair of actions.

Coordination Games; The Prisoner’s Dilemma Tim and Tom are in police custody. Each can confess (C) to a crime or stay silent (S). Confession by both results in 5 years each in jail. Silence by both results in 2 years each in jail. If Tim confesses and Tom stays silent then Tim gets no penalty and Tom gets 10 years in jail (and conversely).

Coordination Games; The Prisoner’s Dilemma -10,0-2,-2 -5,-50,-10 Confess Silent Tom Tim Silent Confess For Tim, Confess strictly dominates Silent.

Coordination Games; The Prisoner’s Dilemma Confess Silent Tom Tim Silent Confess For Tim, Confess strictly dominates Silent. For Tom, Confess strictly dominates Silent. -10,0-2,-2 -5,-50,-10

Coordination Games; The Prisoner’s Dilemma Confess Silent Tom Tim Silent Confess For Tim, Confess strictly dominates Silent. For Tom, Confess strictly dominates Silent. The only NE is (Confess, Confess). -10,0-2,-2 -5,-50,-10

Coordination Games; The Prisoner’s Dilemma Confess Silent Tom Tim Silent Confess For Tim, Confess strictly dominates Silent. For Tom, Confess strictly dominates Silent. The only NE is (Confess, Confess). -10,0-2,-2 -5,-50,-10 But (Silence, Silence) is better for both Tim and Tom.

Coordination Games; The Prisoner’s Dilemma Confess Silent Tom Tim Silent Confess Possible means include future punishments or enforceable contracts. -10,0-2,-2 -5,-50,-10 What is needed is a means of rationally assuring commitment by both players to the most beneficial coordinated actions.

Coordination Games; Assurance Games A simultaneous play game with two “coordinated” NE, one of which gives strictly greater payoffs to each player than does the other. The question is: How can each player give the other an “assurance” that will cause the better NE to be the outcome of the game?

Coordination Games; Assurance Games A common example is the “arms race” problem. India and Pakistan can both increase their stockpiles of nuclear weapons. This is very costly. Having nuclear superiority over the other gives a higher payoff, but the worst payoff to the other. Not increasing the stockpile is best for both.

Coordination Games; Assurance Games Stockpile Don’t Pakistan India Don’t Stockpile 1,45,5 3,34,1

Coordination Games; Assurance Games Stockpile Don’t Pakistan India Don’t Stockpile 1,45,5 3,34,1 The game’s NE are ??

Coordination Games; Assurance Games Stockpile Don’t Pakistan India Don’t Stockpile 1,45,55,5 3,33,34,1 The game’s NE are (Don’t, Don’t) and (Stockpile, Stockpile). Which is the “likely” NE?

Coordination Games; Assurance Games Stockpile Don’t Pakistan India Don’t Stockpile 1,45,55,5 3,33,34,1 The game’s NE are (Don’t, Don’t) and (Stockpile, Stockpile). Which is the “likely” NE? What if India moved first? What action would it choose? Wouldn’t Don’t be best?

Coordination Games; Chicken A simultaneous play game with two “coordinated” NE in which each player chooses the action that is not the action chosen by the other player.

Coordination Games; Assurance Games Two drivers race their cars at each other. A driver who swerves is a “wimp”. A driver who does not swerve is “macho.” If both do not swerve there is a crash and a very low payoff to both. If both swerve then there is no crash and a moderate payoff to both. If one swerves and the other does not then the swerver gets a low payoff and the non- swerver gets a high payoff.

Coordination Games; Assurance Games No Swerve Swerve Dumber Dumb Swerve No Swerve -2,41,1 -5,-54,-2 The game’s NE are ??

Coordination Games; Assurance Games No Swerve Swerve Dumber Dumb Swerve No Swerve -2,41,1 -5,-54,-2 The game’s pure strategy NE are (Swerve, No Swerve) and (No Swerve, Swerve). There is also a mixed strategy NE in which each chooses Swerve with probability ½.

Coordination Games; Assurance Games No Swerve Swerve Dumber Dumb Swerve No Swerve -2,41,1 -5,-54,-2 The game’s pure strategy NE are (Swerve, No Swerve) and (No Swerve, Swerve). There is also a mixed strategy NE in which each chooses Swerve with probability ½. Can Dumb assure himself of a payoff of 4? Only by convincing Dumber that Dumb really will choose No Swerve. What will be convincing?

Some Important Types of Games Games of coordination Games of competition Games of coexistence Games of commitment Bargaining games

Games of Competition Simultaneous play games in which any increase in the payoff to one player is exactly the decrease in the payoff to the other player. These games are thus often called “constant (payoff) sum” games.

Games of Competition D L 2 1 U R 2,-20,0 1,-1 x,-x An example is the game below. What NE can such a game possess?

Games of Competition D L 2 1 U R 2,-20,0 1,-1 x,-x An example is the game below. What NE can such a game possess? If x < 0 then Up ??

Games of Competition D L 2 1 U R 2,-20,0 1,-1 x,-x An example is the game below. What NE can such a game possess? If x < 0 then Up dominates Down.

Games of Competition D L 2 1 U R 2,-20,0 1,-1 x,-x An example is the game below. What NE can such a game possess? If x < 0 then Up dominates Down. If x < 1 then Left ??

Games of Competition D L 2 1 U R 2,-20,0 1,-1 x,-x An example is the game below. What NE can such a game possess? If x < 0 then Up dominates Down. If x < 1 then Left dominates Right.

Games of Competition D L 2 1 U R 2,-20,0 1,-1 x,-x An example is the game below. What NE can such a game possess? If x < 0 then Up dominates Down. If x < 1 then Left dominates Right. Therefore, if x < 0 the NE is ??

Games of Competition D L 2 1 U R 2,-20,0 1,-1 x,-x An example is the game below. What NE can such a game possess? If x < 0 then Up dominates Down. If x < 1 then Left dominates Right. Therefore, if x < 0 the NE is (Up, Left)

Games of Competition D L 2 1 U R 2,-20,0 1,-1 x,-x An example is the game below. What NE can such a game possess? If x < 0 then Up dominates Down. If x < 1 then Left dominates Right. Therefore, if x < 0 the NE is (Up, Left) and if 0 < x < 1 the NE is ??

Games of Competition D L 2 1 U R 2,-20,0 1,-1 x,-x An example is the game below. What NE can such a game possess? If x < 0 then Up dominates Down. If x < 1 then Left dominates Right. Therefore, if x < 0 the NE is (Up, Left) and if 0 < x < 1 the NE is (Down, Left).

Games of Competition D L 2 1 U R 2,-20,0 1,-1 x,-x An example is the game below. What NE can such a game possess? If x 1 then ??

Games of Competition D L 2 1 U R 2,-20,0 1,-1 x,-x An example is the game below. What NE can such a game possess? If x 1 then there is no NE in pure strategies. Is there a mixed-strategy NE?

Games of Competition D L 2 1 U R 2,-20,0 1,-1 x,-x The probability that 2 chooses Left is  L. The probability that 1 chooses Up is  U. x > 1.

Games of Competition D L 2 1 U R 2,-20,0 1,-1 x,-x The probability that 2 chooses Left is  L. The probability that 1 chooses Up is  U. x > 1. EV 1 (U) = 2(1 -  L ). EV 1 (D) = x  L  L.

Games of Competition D L 2 1 U R 2,-20,0 1,-1 x,-x The probability that 2 chooses Left is  L. The probability that 1 chooses Up is  U. x > 1. EV 1 (U) = 2(1 -  L ). EV 1 (D) = x  L  L. >=<>=<  l 1 + (x - 1)  L as  L 1/(1 + x). >=<>=<

Games of Competition D L 2 1 U R 2,-20,0 1,-1 x,-x The probability that 2 chooses Left is  L. The probability that 1 chooses Up is  U. x > 1. EV 1 (U) = 2(1 -  L ). EV 1 (D) = x  L  L. >=<>=<  l 1 + (x - 1)  L as  L 1/(1 + x). >=<>=< EV 2 (L) = - x(1 -  U ). EV 2 (R) = - 2  U - (1 -  U ).

Games of Competition D L 2 1 U R 2,-20,0 1,-1 x,-x The probability that 2 chooses Left is  L. The probability that 1 chooses Up is  U. x > 1. EV 1 (U) = 2(1 -  L ). EV 1 (D) = x  L  L. >=<>=<  l 1 + (x - 1)  L as  L 1/(1 + x). >=<>=< EV 2 (L) = - x(1 -  U ). EV 2 (R) = - 2  U - (1 -  U ). >=<>=< - x + x  U  U as (x – 1)/(1 + x)  U. >=<>=<

Games of Competition D L 2 1 U R 2,-20,0 1,-1 x,-x 1 chooses Up if  L > 1/(1 + x) and Down if  L < 1/(1 + x). 2 chooses Left if  U (x – 1)/(1 + x).

Games of Competition 1: Up if  L > 1/(1 + x); Down if  L < 1/(1 + x). 2: Left if  U (x – 1)/(1 + x). UU UU LL LL

Games of Competition 1: Up if  L > 1/(1 + x); Down if  L < 1/(1 + x). 2: Left if  U (x – 1)/(1 + x). UU UU LL LL /(1+x)(x-1)/(1+x)

Games of Competition 1: Up if  L > 1/(1 + x); Down if  L < 1/(1 + x). 2: Left if  U (x – 1)/(1 + x). UU UU LL LL /(1+x)(x-1)/(1+x)

Games of Competition 1: Up if  L > 1/(1 + x); Down if  L < 1/(1 + x). 2: Left if  U (x – 1)/(1 + x). UU UU LL LL /(1+x)(x-1)/(1+x)

Games of Competition 1: Up if  L > 1/(1 + x); Down if  L < 1/(1 + x). 2: Left if  U (x – 1)/(1 + x). UU LL /(1+x) UU LL (x-1)/(1+x)

Games of Competition 1: Up if  L > 1/(1 + x); Down if  L < 1/(1 + x). 2: Left if  U (x – 1)/(1 + x). UU LL /(1+x) UU LL (x-1)/(1+x)

Games of Competition 1: Up if  L > 1/(1 + x); Down if  L < 1/(1 + x). 2: Left if  U (x – 1)/(1 + x) UU LL LL 1/(1+x) UU (x-1)/(1+x)

Games of Competition 1: Up if  L > 1/(1 + x); Down if  L < 1/(1 + x). 2: Left if  U (x – 1)/(1 + x). 1 UU LL (x-1)/(1+x) 1/(1+x) When x > 1 there is only a mixed-strategy NE in which 1 plays Up with probability (x – 1)/(x + 1) and 2 plays Left with probability 1/(1 + x).

Some Important Types of Games Games of coordination Games of competition Games of coexistence Games of commitment Bargaining games

Coexistence Games Simultaneous play games that can be used to model how members of a species act towards each other. An important example is the hawk-dove game.

Coexistence Games; The Hawk-Dove Game “Hawk” means “be aggressive.” “Dove” means “don’t be aggressive.” Two bears come to a fishing spot. Either bear can fight the other to try to drive it away to get more fish for itself but suffer battle injuries, or it can tolerate the presence of the other, share the fishing, and avoid injury.

Coexistence Games; The Hawk-Dove Game 8,0 4,40,8 Hawk Dove HawkDove Bear 2 Bear 1 Are there NE in pure strategies? -5,-5

Coexistence Games; The Hawk-Dove Game 8,0 4,40,8 Hawk Dove HawkDove Bear 2 Bear 1 Are there NE in pure strategies? Yes (Hawk, Dove) and (Dove, Hawk). Notice that purely peaceful coexistence is not a NE. -5,-5

Coexistence Games; The Hawk-Dove Game 8,0 4,40,8 Hawk Dove HawkDove Bear 2 Bear 1 Is there a NE in mixed strategies? -5,-5

Coexistence Games; The Hawk-Dove Game 8,0 4,40,8 Hawk Dove HawkDove Bear 2 Bear 1 Is there a NE in mixed strategies?  1 H is the prob. that 1 chooses Hawk.  2 H is the prob. that 2 chooses Hawk. What are the players’ best-response functions? -5,-5

Coexistence Games; The Hawk-Dove Game 8,0 4,40,8 Hawk Dove HawkDove Bear 2 Bear 1  1 H is the prob. that 1 chooses Hawk.  2 H is the prob. that 2 chooses Hawk. What are the players’ best-response functions? EV 1 (H) = -5  2 H + 8(1 -  2 H ) =  2 H. EV 1 (D) =  2 H. -5,-5

Coexistence Games; The Hawk-Dove Game 8,0 4,40,8 Hawk Dove HawkDove Bear 2 Bear 1  1 H is the prob. that 1 chooses Hawk.  2 H is the prob. that 2 chooses Hawk. What are the players’ best-response functions? EV 1 (H) = -5  2 H + 8(1 -  2 H ) =  2 H. EV 1 (D) =  2 H  2 H  2 H as  2 H 4/9. >=<>=< <=><=> -5,-5

Coexistence Games; The Hawk-Dove Game 8,0 4,40,8 Hawk Dove HawkDove Bear 2 Bear 1  1 H is the prob. that 1 chooses Hawk.  2 H is the prob. that 2 chooses Hawk. Bear 1 1H1H 2H2H /9 -5,-5 EV 1 (H) = -5  2 H + 8(1 -  2 H ) =  2 H. EV 1 (D) =  2 H  2 H  2 H as  2 H 4/9. >=<>=< <=><=>

Coexistence Games; The Hawk-Dove Game Bear 1Bear 2 2H2H 1H1H /9 1H1H 2H2H

Coexistence Games; The Hawk-Dove Game Bear 1Bear 2 2H2H 1H1H /9 1H1H 2H2H

Coexistence Games; The Hawk-Dove Game Bear 1Bear 2 1H1H 2H2H /9 1H1H 2H2H

Coexistence Games; The Hawk-Dove Game 1H1H 2H2H /9 The game has a NE in mixed-strategies in which each bear plays Hawk with probability 4/9.

Coexistence Games; The Hawk-Dove Game 8,0-5,-5 4,40,8 Hawk Dove HawkDove Bear 2 Bear 1 For each bear, the expected value of the mixed-strategy NE is (-5)× + 8× + 4× =, a value between -5 and +4. Is this NE focal? / / / /

Some Important Types of Games Games of coordination Games of competition Games of coexistence Games of commitment Bargaining games

Commitment Games Sequential play games in which One player chooses an action before the other player chooses an action. The first player’s action is both irreversible and observable by the second player. The first player knows that his action is seen by the second player.

Commitment Games 5,9 5,5 7,6 5, a b e c d f Game Tree Player 1 has two actions, a and b. Player 2 has two actions, c and d, following a, and two actions e and f following b. Player 1 chooses his action before Player 2 chooses her action. Direction of play

Commitment Games 5,9 5,5 7,6 5, a b e c d f Is a claim by Player 2 that she will commit to choosing action c if Player 1 chooses a credible to Player 1?

Commitment Games 5,9 5,5 7,6 5, a b e c d f Is a claim by Player 2 that she will commit to choosing action c if Player 1 chooses a credible to Player 1? Yes.

Commitment Games 5,9 5,5 7,6 5, a b e c d f Is a claim by Player 2 that she will commit to choosing action c if Player 1 chooses a credible to Player 1? Yes. Is a claim by Player 2 that she will commit to choosing action e if Player 1 chooses b credible to Player 1?

Commitment Games 5,9 5,5 7,6 5, a b e c d f Is a claim by Player 2 that she will commit to choosing action c if Player 1 chooses a credible to Player 1? Yes. Is a claim by Player 2 that she will commit to choosing action e if Player 1 chooses b credible to Player 1? Yes.

Commitment Games 5,95,9 5,5 7,67,6 5, a b e c d f So Player 1 should choose action ??

Commitment Games 5,9 5,5 7,67,6 5, a b e c d f So Player 1 should choose action b.

Commitment Games 5,3 5,5 7,6 5, a b e c d f Change the game.

Commitment Games 5,3 5,5 7,6 5, a b e c d f Is a claim by Player 2 that she will commit to choosing action c if Player 1 chooses a credible to Player 1?

Commitment Games 5,3 5,55,5 7,67,6 5, a b e c d f Is a claim by Player 2 that she will commit to choosing action c if Player 1 chooses a credible to Player 1? No. If Player 1 chooses action a then Player 2 does best by choosing action d. What should Player 1 do?

Commitment Games 5,3 5,55,5 7,67,6 5, a b e c d f Is a claim by Player 2 that she will commit to choosing action c if Player 1 chooses a credible to Player 1? No. If Player 1 chooses action a then Player 2 does best by choosing action d. What should Player 1 do? Still choose b.

Commitment Games 5,3 5,5 7,3 5, a b e c d f Change the game. 5,9 15,5

Commitment Games 5,9 7,3 5, a b e c d f Can Player 1 get 15 points? 15,5

Commitment Games 5,95,9 7,3 5, a b e c d f Can Player 1 get 15 points? If Player 1 chooses a then Player 2 will choose c and Player 1 will get only 5 points. 15,5

Commitment Games 5,95,9 7,3 5, a b e c d f Can Player 1 get 15 points? If Player 1 chooses a then Player 2 will choose c and Player 1 will get only 5 points. If Player 1 chooses b then Player 2 will choose f and again Player 1 will get only 5 points. 15,5

Commitment Games 5,95,9 7,3 5, a b e c d f If Player 1 can change payoffs so that a commitment by Player 2 to choose d after a is credible then Player 1’s payoff rises from 5 to 15, a gain of ,5

Commitment Games 5,95,9 10,10 7,3 5, a b e c d f If Player 1 can change payoffs so that a commitment by Player 2 to choose d after a is credible then Player 1’s payoff rises from 5 to 15, a gain of 10. If Player 1 gives 5 of these points to Player 2 then Player 2’s commitment is credible. Player 1 cannot get 15 points.

Commitment Games 5,9 10,10 7,3 5, a b e c d f Credible NE of this type are called subgame perfect. What exactly is this game’s SPE? It insists that every action chosen is rational for the player who chooses it.

Commitment Games 5,9 10,10 7,3 5, a b e c d f Credible NE of this type are called subgame perfect. What exactly is this game’s SPE? It insists that every action chosen is rational for the player who chooses it.

Commitment Games 5,9 10,10 7,3 5, a b e c d f Credible NE of this type are called subgame perfect. What exactly is this game’s SPE? It insists that every action chosen is rational for the player who chooses it.

Commitment Games 5,9 10,10 7,3 5, a b e c d f Credible NE of this type are called subgame perfect. What exactly is this game’s SPE? It insists that every action chosen is rational for the player who chooses it.

Commitment Games 5,9 10,10 7,3 5, a b e c d f Credible NE of this type are called subgame perfect. What exactly is this game’s SPE? It insists that every action chosen is rational for the player who chooses it.

Some Important Types of Games Games of coordination Games of competition Games of coexistence Games of commitment Bargaining games

Bargaining Games Two players bargain over the division of a pie of size 1. What will be the outcome? Two approaches: Nash’s axiomatic bargaining. Rubinstein’s strategic bargaining.

Strategic Bargaining The players have 3 periods in which to decide how to divide the pie; else both get nothing. Player A discounts next period’s payoffs by . Player B discounts next period’s payoffs by . The players alternate in making offers, with Player A starting in period 1. If the player who receives an offer accepts it then the game ends immediately. Else the game continues to the next period.

Strategic Bargaining 0 1 A 0 1 B 0 1 A B B A x1x1 (x 1,1-x 1 ) Y N x2x2 x3x3 (x 3,1-x 3 ) (x 2,1-x 2 ) (0,0) Y Y N N Period 1: A offers x 1. B responds. Period 2: B offers x 2. A responds. Period 3: A offers x 3. B responds.

Strategic Bargaining 0 1 A B x3x3 Y N Period 3: A offers x 3. B responds. How should B respond to x 3 ? (x 3,1-x 3 ) (0,0)

Strategic Bargaining 0 1 A B x3x3 Y N Period 3: A offers x 3. B responds. How should B respond to x 3 ? Accept if 1 – x 3 ≥ 0; i.e., accept any x 3 ≤ 1. (x 3,1-x 3 ) (0,0)

Strategic Bargaining 0 1 A B x3x3 (0,0) Y N Period 3: A offers x 3. B responds. How should B respond to x 3 ? Accept if 1 – x 3 ≥ 0; i.e., accept any x 3 ≤ 1. Knowing this, what should A offer? (x 3,1-x 3 )

x 3 =1 Strategic Bargaining 0 A B (1,0) (0,0) Y N Period 3: A offers x 3 = 1. B accepts. How should B respond to x 3 ? Accept if 1 – x 3 ≥ 0; i.e., accept any x 3 ≤ 1. Knowing this, what should A offer? x 3 = 1.

x 3 =1 Strategic Bargaining 0 1 A 0 1 B 0 A B A x1x1 (x 1,1-x 1 ) Y N x2x2 (x 2,1-x 2 ) Y N Period 1: A offers x 1. B responds. Period 2: B offers x 2. A responds. B Y N Period 3: A offers x 3 = 1. B accepts. (1,0) (0,0)

x 3 =1 Strategic Bargaining 0 1 B 0 A A x2x2 (x 2,1-x 2 ) Y N Period 2: B offers x 2. A responds. B Y N Period 3: A offers x 3 = 1. B accepts. In Period 3 A gets a payoff of 1. In period 2, when replying to B’s offer of x 2, the present-value to A of N is thus ?? (1,0) (0,0)

x 3 =1 Strategic Bargaining 0 1 B 0 A A x2x2 (x 2,1-x 2 ) Y N Period 2: B offers x 2. A responds. B Y N Period 3: A offers x 3 = 1. B accepts. In Period 3 A gets a payoff of 1. In period 2, when replying to B’s offer of x 2, the present-value to A of N is thus . (1,0) (0,0)

Strategic Bargaining 0 1 B A x2x2 (x 2,1-x 2 ) Y N Period 2: B offers x 2. A responds. In Period 3 A gets a payoff of 1. In period 2, when replying to B’s offer of x 2, the present-value to A of N is thus . What is the most B should offer to A?

Strategic Bargaining 0 1 B A x2=x2= ( ,1-  ) Y N Period 2: B offers x 2 = . A accepts. In Period 3 A gets a payoff of 1. In period 2, when replying to B’s offer of x 2, the present-value to A of N is thus . What is the most B should offer to A? x 2 = .

x 3 =1 Strategic Bargaining 0 1 A 0 1 B 0 A B A x1x1 (x 1,1-x 1 ) Y N Y N Period 1: A offers x 1. B responds. B Y N Period 3: A offers x 3 = 1. B accepts. Period 2: B offers x 2 = . A accepts. x2=x2= ( ,1-  ) (1,0) (0,0)

Strategic Bargaining 0 1 A 0 1 B B A x1x1 (x 1,1-x 1 ) Y N Y N Period 1: A offers x 1. B responds. Period 2: B offers x 2 = . A accepts. x2=x2= ( ,1-  ) In period 2 A will accept . Thus B will get the payoff 1 -  in period 2. What is the present- value to B in period 1 of N ?

Strategic Bargaining 0 1 A 0 1 B B A x1x1 (x 1,1-x 1 ) Y N Y N Period 1: A offers x 1. B responds. Period 2: B offers x 2 = . A accepts. x2=x2= ( ,1-  ) In period 2 A will accept . Thus B will get the payoff 1 -  in period 2. What is the present- value to B in period 1 of N ?  (1 -  ).

Strategic Bargaining 0 1 A 0 1 B B A x1x1 (x 1,1-x 1 ) Y N Y N Period 1: A offers x 1. B responds. Period 2: B offers x 2 = . A accepts. x2=x2= ( ,1-  ) In period 2 A will accept . Thus B will get the payoff 1 -  in period 2. What is the present- value to B in period 1 of N ?  (1 -  ). What is the most that A should offer to B in period 1?

Strategic Bargaining 0 1 A 0 1 B B A x1x1 (1-  (1 -  ),  (1 -  )) Y N Y N Period 1: A offers x 1. B responds. Period 2: B offers x 2 = . A accepts. x2=x2= ( ,1-  ) In period 2 A will accept . Thus B will get the payoff 1 -  in period 2. What is the present- value to B in period 1 of N ?  (1 -  ). What is the most that A should offer to B in period 1? 1 – x 1 =  (1 -  ); i.e. x 1 = 1 -  (1 -  ). B will accept.

x 3 =1 Strategic Bargaining 0 1 A 0 1 B 0 A B A Y N Y N Period 1: A offers x 1 = 1-  (1 -  ). B accepts. B Y N Period 3: A offers x 3 = 1. B accepts. Period 2: B offers x 2 = . A accepts. x2=x2= ( ,1-  ) (1-  (1 -  ),  (1 -  )) x 1 =1-  (1-  ) (1,0) (0,0)

Strategic Bargaining Notice that the game ends immediately, in period 1. Player A gets 1 -  (1 –  ) units of the pie. Player B gets  (1 –  ) units. Which is the larger? x 1 = 1 -  (1 –  ) ≥ ½   ≤ 1/2(1 -  ) so Player A gets more than Player B if Player B is “too impatient” relative to Player A.

Strategic Bargaining Suppose the game is allowed to continue forever (infinitely many periods). Then using the same reasoning shows that the subgame perfect equilibrium results in Players 1 and 2 respectively getting and pie units. Player 1’s share rises as  and . Player 2’s share rises as  and .