Problem 4. This is an extension of the famous Matching Pennies game that we saw

Question

Problem 4. This is an extension of the famous Matching Pennies game that we saw in class. There are two players, Anne and Bill. Each of them chooses to display, at a given signal and simultaneously, either a one cent coin (1c), a five cent coin (5c), or a ten cent coin (10c). If the sum of the displayed coins turns out to be odd, then Anne wins the coin that Bill displayed Ot osen coin herwise, Bill wins Anne's ch (a) Write down the strategies of both Anne and Bill, and represent the game in its matrix form, carefully labeling payoffs for Anne (row player) and Bill (column player). As payoffs for both players, you can consider just the amount earned or lost as a result of [4 points] the game. (b) Can you find a pure strategy Nash equilibrium? [4 points] (c) If the answer to the above question is negative, can you find a mixed strategy Naslh [2 points] Hint: consider Anne's best response to Bill's mixed strategy in which he only puts positive probability to 1c and 10c. Do the same for Bill. Verify that the strategy profile you found this equilibrium instead? way is a N

Explanation / Answer

Answer

a)

The strategies available to Anne are 1 cent coin, 5 cent coin and 10 cent coin. Similarly strategies available to Bill are 1 cent coin, 5 cent coin and 10 cent coin. Combination of strategies and the respective payoff are given in the following Payoff table.

Normal form of the game is as follows:

Bill

b)

There is No Pure strategy Nash equilibrium, Because there is no such strategy exist such that a Anne will choose given Strategy of other.

We can see from the table that:

When Anne Chooses 1 cent, Bill can choose any either 1 cent or 5 cent. If Bill chooses 1 or 5 cent then Anne will choose 10 Cent. hence When Anne chooses 1 cent there is no nash equilibrium (pure Strategy)

When Anne Chooses 5 cent, Bill can choose any either 1 cent or 5 cent. If Bill chooses 1 or 5 cent then Anne will choose 10 Cent. hence When Anne chooses 5 cent there is no nash equilibrium (pure Strategy).

When Anne Chooses 10 cent, Bill can choose 10 cent. If Bill chooses 10 cent then Anne will choose either 1Cent or 5 cent. hence When Anne chooses 10 cent there is no nash equilibrium (pure Strategy).

HENCE THERE IS NO PURE STRATEGY NASH EQUILIBRIUM.

c)

Now Note that For Bill, Strategy 1 cent strictly dominates strategy 5 cent. Hence Bill will never choose 5 cent. Similarly, Note that For Anne, Strategy 1 cent strictly dominates strategy 5 cent. Hence Bill will never choose 5 cent

Suppose Bill chooses 1cent coin with probability p  and hence he will choose 10 cent with probability 1 - p and 5 cent with 0 probability.

Suppose Anne chooses 1cent coin with probability q  and hence he will choose 10 cent with probability 1 - q and 5 cent with 0 probability

We have to find q1 such that whatever Bill Choose Anne will get the same expected payoff from all strategies Suppose Bill chooses 1 Cent. Anne expected value = -q + 1 -q = 1-2q.

Suppose Bill chooses 10 Cent. Anne expected value = -10 +20q

=> -10 +20q = 1-2q. Hence q = 11/22 = 0.5.

Now,

We have to find p such that whatever Bill Choose Anne will get the same expected payoff from all strategies. Suppose Anne chooses 1 Cent. Bill expected value = p - 10(1 - p)=11p - 10.

Suppose Anne chooses 10 Cent. Bill expected value = -p +10(1-p) = 10 - 11p

=> 10 - 11p = 11p - 10. Hence p = 20/22 = 10/11.

So Mixed Strategy nash Equilibrium is When Anne chooses (1,5,10) with probability(1/0 , 0 , 1/2) and Bill chooses (1,5,10) with Probability (10/11 , 0 , 1/11)

1 cent coin 5 cent coin 10 cent coin 1 cent coin -1 , +1 -1 , +1 +10 , -10 Anne 5 cent coin -5 , +5 -5 , +5 +10 , -10 10 cent coin +1 , -1 +5 , -5 -10 , +10

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