Two-Player Zero-Sum Games

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Presentation transcript:

Two-Player Zero-Sum Games Finding the saddle-point – Example #2

X Y A -3 2 B 4 -1 Finding Saddle Points Consider the following 2-player zero-sum game… Player 2 X Y A -3 2 B 4 -1 Player 1

X Y A -3 2 B 4 -1 Finding Saddle Points Consider the following 2-player zero-sum game… Player 2 X Y A -3 2 B 4 -1 First, determine if saddle-point is in pure strategies…. we determine minimax and maximin strategies. Player 1

X Y A -3 2 B 4 -1 Finding Saddle Points Consider the following 2-player zero-sum game… P2 X Y A -3 2 B 4 -1 maximum of minimum values -3 -1 P1 4 2 minimum of maximum values

X Y A -3 2 B 4 -1 Finding Saddle Points Consider the following 2-player zero-sum game… P2 X Y A -3 2 B 4 -1 B is the maximin strategy for player 1 -3 -1 P1 4 2 minimum of maximum values

X Y A -3 2 B 4 -1 Finding Saddle Points Consider the following 2-player zero-sum game… P2 X Y A -3 2 B 4 -1 B is the maximin strategy for player 1 -3 -1 P1 4 2 Y is the minimax strategy for player 2

X Y A -3 2 B 4 -1 Finding Saddle Points Consider the following 2-player zero-sum game… P2 X Y A -3 2 B 4 -1 B is the maximin strategy for player 1 -3 -1 P1 4 2 Y is the minimax strategy for player 2

X Y A -3 2 B 4 -1 Finding Saddle Points -3 -1 4 2 P2 Maximin strategy for player 1 -3 -1 P1 4 2 Minimax strategy for player 2 If player 1 chooses strategy A, then player 2’s best strategy results in the payoff -3 If player 1 chooses strategy B, then player 2’s best strategy results in the payoff -1 Therefore, assuming player 2 is always playing his best strategy for any choice by player 1, the best strategy for player 1 is B… this is called the maximin strategy.

X Y A -3 2 B 4 -1 Finding Saddle Points -3 -1 4 2 P2 Maximin strategy for player 1 -3 -1 P1 4 2 Minimax strategy for player 2 If player 2 chooses X then the best strategy for player 1 results in 4. If player 2 chooses Y then the best strategy for player 1 results in 2. Therefore, assuming player 1 is always playing his best strategy for any strategy choice of player 2, the best strategy for player 2 is Y. This is the minimax strategy for player 2.

X Y A -3 2 B 4 -1 Finding Saddle Points -3 -1 4 2 P2 P1 These values are not the same and therefore the optimal strategies – the saddle point – lies in mixed strategies for each player. 4 2 Assuming that player 2 is playing his best, player 1 could be sure that the worst that could happen would be to get a payoff of -1 if he continued with strategy B and it might turn out better, if player 2 mistakenly chose X.

X Y A -3 2 B 4 -1 Finding Saddle Points -3 -1 4 2 P2 Assuming that player 1 is playing his best, player 2 could choose strategy Y and the worst that could happen is to get a 2 (and it could turn out better for player 2, if player 1 mistakenly chose B.) -3 -1 P1 4 2 The situation is that an optimal strategy for each will be mixed strategies because, the worst that could happen to each is not the same value… that is to say, based on the maximin and minimax strategies of each, each knows the worst that could happen assuming the other plays their best. Each player will try to do better than that by mixing strategies on repeated plays of the game.

X Y A -3 2 B 4 -1 Finding Saddle Points Assign probabilities for repeated play for each strategy. P2 p 1-p X Y A -3 2 B 4 -1 q 1-q P1 First, let’s determine P2’s optimal mixed strategy…To do this, we will consider the payoffs that P1 would receive with pure strategies corresponding to P2’s mixed strategy… EA= -3p + 2(1-p) = -3p+2-2p = -5p+2 EB = 4p + -1(1-p) = 4p -1+p = 5p-1

Graphing the Payoff Functions Find intersection by solving equations… 5p-1 = -5p+2 10p = 3 p=3/10 then plug p = 3/10 into either function to get payoff of 1/2 expected value Intersection at (3/10, 1/2) 1/2 represents the expected payoff to the row player (player 1) and is therefore the referred to as the value of the game. Note that the expected payoff to the column player (player 2) would therefore by -1/2). EB = 5p - 1 4 1 p -1 1 -3 EA = -5p + 2

X Y A -3 2 B 4 -1 Finding Saddle Points P2 p 1-p q 1-q P1 Now, for player 1… we assume pure strategies for player 2 and find q and 1-q which represent the optimal mixed strategy for player 1… EX= -3q + 4(1-q) = -3q + 4 – 4q = -7q + 4 EY = 2q -1(1-q) = 2q – 1 + q = 3q - 1

Graphing the Payoff Functions Find intersection by solving equations… -7q + 4 = 3q -1 10q = 5 q = 1/2 The goal for player 1 is to maximize his own payoff. Because it is a zero sum game, that’s the same as minimizing the other player’s payoff. expected value Intersection at (1/2, 1/2) For example, if player 1 chose a value of q = 1 which corresponds to playing strategy A, then the best strategy for player 2 is to pick strategy X. Player 1 finds that q=1/2 is optimal because player 2 can not get a lower payoff at this point. 4 EY = 3q -1 1 q -1 1 -3 EX = -7q + 4

Graphing the Payoff Functions Find intersection by solving equations… -7q + 4 = 3q -1 10q = 5 q = 1/2 The conclusion is that an optimal mixed strategy for player 1 is play strategy A with probability 1/2 and strategy B with probability 1/2. expected value Intersection at (1/2, 1/2) 4 EY = 3q -1 1 q -1 1 -3 EX = -7q + 4

X Y A -3 2 B 4 -1 Writing the Answer We were seeking an optimal strategy for each player in this 2-player zero-sum game. The answer is that the optimal strategies for each player lie in mixed strategies. We’ve found that the optimal strategy for player 2 is play strategy X with probability 3/10 and therefore play strategy Y with probability 7/10. And we found that the optimal strategy for player 1 is to play strategy A with probability 1/2 and strategy B with probability 1/2. Player 2 X Y A -3 2 B 4 -1 Player 1 We’ve also found that the value of the same is 1/2. Note that the value of the game is the payoff to the row player (and thus negating the value of the game is the payoff to the column player). Also note that the value of the game is always the second coordinate in both calculations of an intersection point for the payoff functions.

X Y A -3 2 B 4 -1 Writing the Answer We were seeking an optimal strategy for each player in this 2-player zero-sum game. Player 2 Here is a succinct way to write the final answer…. Optimal strategies: Player 2: (3/10, 7/10) Player 1: (1/2, 1/2) Value of the game: 1/2 X Y A -3 2 B 4 -1 Player 1 In writing it this way, it is understood that the first coordinate is the probability for playing the first strategy and the second coordinate is the probability for playing the other strategy when reading the strategies from the given table in the natural order – from left to right and top to bottom.