2. (50) So far, we have considered only what are known as pure strategies, where
ID: 1127830 • Letter: 2
Question
2. (50) So far, we have considered only what are known as pure strategies, where players simply choose one of the actions available to them. However, employing pure strategies also makes the player's actions predictable and easily countered. Indeed, there is no guarantee that a game must have a Nash equilibrium in pure strategies, as we can imagine many that will inevitably result in a endless sequence of counter moves. In this problem, we will investigate one such game, and nd a Nash equilibrium by expanding the scope of the player's choices beyond only pure strategies.
Suppose that two players, Abe and Liz, are playing a game known as matching pennies, in which both simul-taneously reveal a penny as either heads H or tails T . Abe wins $1 from Liz if both players choose the same side, while Liz wins $1 from Abe if they choose di erent sides.
(a) (10) Write the payo matrix for this game and show that none of the four pure strategy outcomes is a Nash equilibrium.
Matching pennies is an example of a game that has no Nash equilibria in pure strategies, as if both players commit to a single action, one of them will always want to switch. However, a Nash equilibrium does in fact exist in this game, but to nd it we consider the players employing mixed strategies, where they choose their actions randomly rather than deciding on one or the other with certainty.
Suppose that Abe, frustrated with his inability to win consistently picking either heads or tails, decides that rather than picking one or the other, he will just ip the coin and play whatever it lands on. That is, he randomly chooses his action, choosing either heads or tails, each with probability 12 .
(b) (10) Liz observes Abe's novel new strategy and thinks about how best to respond to it. What is her average payo if she plays heads? What if she plays tails?
(c) (10) Liz resolves that the best way to deal with Abe's mixed strategy is to employ a mixed strategy of her own, playing heads with probability 23 and tails with probability 13 . What is Liz's expected payo ? Does it make sense for her to employ the mixed strategy?
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(d) (10) Abe now observes Liz's mixed strategy. What is his average payo if he plays heads? What if he plays tails? Does it makes sense for him to continue randomizing between heads and tails?
(e) (10) Find the mixed strategy Nash equilibrium in the matching pennies game.
Explanation / Answer
(a)
Thus, there is no pure strategy nash equilibrium.
(b) If Liz plays heads, expected payoff
= probability that Abe plays heads * payoff of Liz for same sides + probability that Abe plays tails * payoff of Liz for different sides
= 1/2 * -1 + 1/2 * 1
= 0
If Liz plays tails, expected payoff
= probability that Abe plays heads * payoff of Liz for different sides + probability that Abe plays tails * payoff of Liz for same sides
= 1/2 * 1 + 1/2 * -1
= 0
(c) Expected payoff of Liz
= probability that abe plays heads* prob that liz plays heads * payoff of liz for same sides+ probability that abe plays heads* prob that liz plays tails * payoff of liz for different sides + probability that abe plays tails* prob that liz plays tails * payoff of liz for same sides + probability that abe plays tails* prob that liz plays heads * payoff of liz for different sides
= 1/2* 2/3 * -1 + 1/2 * 1/3* 1 + 1/2* 1/3 * -1 + 1/2 * 2/3* 1
= 0
The expected payoff still remains 0. It does'nt make sense for him to employ mixed strategy.
(d)
If Abe plays heads, expected payoff
= prob that liz plays heads * payoff of abe for same sides+ prob that liz plays tails * payoff of abe for different sides
= 2/3 * 1 + 1/3* -1
= 1/3
If Abe plays tails, expected payoff
= prob that liz plays heads * payoff of abe for different sides+ prob that liz plays tails * payoff of abe for same sides
= 2/3 * -1 + 1/3* 1
= -1/3
If he doesn't randomize, expected payoff
= probability that abe plays heads* prob that liz plays heads * payoff of abe for same sides+ probability that abe plays heads* prob that liz plays tails * payoff of abe for different sides + probability that abe plays tails* prob that liz plays heads * payoff of abe for different sides+ probability that abe plays tails* prob that liz plays tails * payoff of abe for samesides
= 1/2* 2/3 * -1 + 1/2 * 1/3* 1 + 1/2* 2/3 * 1 + 1/2 * 1/3* *-1
= 0
Thus, it still makes sense for him to continue randomizing.
I can solve only 4 parts as per Chegg guidelines.
Abe H T Liz H -1,1 1,-1 T 1,-1 -1,1Related Questions
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