Question

Consider a following zero-sum game of two players

a) Reduce the initial game to a 2x2 game. Eliminate only strictly dominated strategies. In the obtained 2x2 game name Player 1’s (Row player’s) strategies “Up” and “Down” and Player 2’s (Column player’s) strategies “Left” and “Right”.

b) Find all Nash equilibria of the 2x2 game (both in pure and mixed strategies)

ALL ANSWERS MUST BE EXPLAINED.

2	0	1	-1
1	0	1	2
3	1	2	0

Answer 1

a) In this game, at first we will eliminate few strategies using the rule of dominance.

For player 1, Row 3 strictly dominates Row 1 as the payoffs from row 1 is lower than the payoffs from Row 3 for every strategy that Player 2 may choose to play. Now, the matrix becomes

1 0 1 2

3 1 2 0

Now if we look it from the perspective of player 2, as it is a zero sum game, all the payoffs will become the negative of what they are. So the matrix for player 2 becomes

-1 0 -1 -2

-3 -1 -2 0

For player 2, column 2 strictly dominates column 1 as all the payoffs of column 2 are higher than the payoffs of column 1 for every straregy that player 1 may choose. Similarly column 2 strictly dominates column 3 as all the payoffs of column 2 are higher than the payoffs of column 3 for every straregy that player 1 may choose.

So, the payoff matrix for player 2 becomes

0 -2

-1 0

The overall payoff matrix will be seen as

Left Right

Up. 0,0 2,-2

Down 1,-1   0,0

Now, we see that if player 1 chooses up, player 2 would choose left.

If player 1 chooses down, player 2 would choose right.

If player 2 chooses left, player 1 would choose down.

If player 2 chooses right, player 1 would choose up.

So, we can see that there is no pure nash equilibrium in this game.

Now, we will look for mixed strategy equilibrium. For player 1, the utility from playing strategy up and strategy down should be equal. Let the probability of playing strategy left for player 2 would be x and the probability of playing strategy right for player 2 will be (1-x).

So, the expected utility from strategy Up for player 1 will be = 0*x + 2*(1-x) = 2-2x

The expected utility from strategy down for player 1 will be = 1*x + 0*(1-x) = x

So, as the expected utility has to be same,

2-2x = x

x = 2/3 = 0.6667.

So the probability that player 2 chooses left is 0.6667 and the probability that player 2 chooses right is 1-0.6667 = 0.33333.

For player 2 , the utility from playing strategy left and strategy right should be equal. Let the probability of playing strategy up  for player 1 would be y and the probability of playing strategy down for player 1 will be (1-y ).

So, the expected utility from strategy Left for player 2 will be = 0*y -1*(1-y ) = y-1

The expected utility from strategy right for player 2 will be = -2*y + 0*(1-y ) = -2y

So, as the expected utility has to be same,

y-1 = -2y

= y = 1/3 = 0.3333

So, the probability with which player 1 chooses up is 0.3333 and the probability with which player 1 chooses down is (1-0.3333) = 0.66667.

So, we can coclude that player 1 plays the mixed strategy of (Up, Down) = (0.3333,0.6667). So, player 1 has an expected payout = 0.6666.

Player 2 plays the mixed strategy of (Left, Right) = (0.6667,0.3333). Player 2 has an expected payout = -0.6667.

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