Lesson overview Chapter 4 Simultaneous Move Games with Pure Strategies I: Discrete Strategies Lesson I.5 Simultaneous Move Theory Each Example Game Introduces some Game Theory • Example 1: A Normal Form • Example 2: Nash Equilibrium • Example 3: Dominate Strategies • Example 4: Iterated Dominance • Example 5: Minimax Lesson I.6 Simultaneous Move Problems Lesson I.7 Simultaneous Move Applications BA 592 Lesson I.5 Simultaneous Move Theory 1

Example 1: A Normal Form Game tables or normal forms condense the information in a game tree or extensive form. Like the extensive form, the normal form specifies strategies for every player and the outcomes of the actions taken by all players. But unlike the extensive form, the normal form does not model the order of the actions. Normal forms are the simplest way to model games where actions are simultaneous.

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Example 1: A Normal Form Walmart and Costco both sell emergency food supplies in a weather-proof bucket that provides 275 delicious easy-to-prepare meals, including potato soup and corn chowder. The unit cost to both retailers is $75. The retailers compete on price: the low price retailer gets all the market and they split the market if they have equal prices. Suppose they consider prices $75, $85, and $95, and suppose market demands at those prices are 140, 100, and 80.

Define the normal form for this Price Competition Game .

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Example 1: A Normal Form To begin, at Walmart price $95 and Costco price $85, Costco gets the entire market demand of 100. Hence, Walmart makes $0 and Costco makes $(85-75)x100 = $1,000.

Walmart $75 $85 $95 $75 0,0 0,0 0,0 Costco $85 0,0 500,500 0,1000 $95 0,0 1000,0 800,800

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Example 2: Nash Equilibrium A Nash equilibrium are strategies for all players so that each player’s strategy is a best response to the strategies of all other players. Put another way, each player’s strategy is a best response given his beliefs about the strategies played by the other players, and those beliefs are, in fact, the actual strategies played by the other players.

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Example 2: Nash Equilibrium Walmart and Costco consider two possibilities: Either continue their price competition as described in Example 1, or offer the following low-price guarantee: We guarantee lower prices than any other store, and we do everything in our power to ensure that you're not paying too much for your purchase. That's why we offer our 110% Low Price Guarantee. If you find a lower advertised price, simply let us know and we'll gladly meet that price and beat it by an additional 10% of the difference!

To decide whether to offer that guarantee, reconsider the normal form for the original Price Competition Game and find the Nash equilibrium. Then, define the normal form for the Price Competition Game modified by the 110% Low Price Guarantee , and find the Nash equilibrium. Compare equilibria.

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Example 2: Nash Equilibrium The normal form for the original Price Competition Game has two Nash equilibria: (Walmart price = $75, Costco price = $75) and (Walmart price = $85, Costco price = $85).

Walmart $75 $85 $95 $75 0,0 0,0 0,0 Costco $85 0,0 500,500 0,1000 $95 0,0 1000,0 800,800

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Example 2: Nash Equilibrium The normal form for the Price Competition Game modified by the 110% Low Price Guarantee has three Nash equilibria: (Walmart price = $75, Costco price = $75) and (Walmart price = $85, Costco price = $85) and (Walmart price = $95, Costco price = $95).

Costco Walmart $75 $85 $95 $75 0,0 0,0 0,0 $85 0,0 500,500 500,500 $95 0,0 500,500 800,800

The new Nash equilibrium (Walmart price = $95, Costco price = $95) is best for both firms and worst for consumers!

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Example 3: Dominate Strategies A dominate strategy for a player gives better payoffs for that player compared with any other strategy, no matter what other players choose for their strategies. A weakly dominate strategy for a player gives at least as good payoffs for that player compared with any other strategy, no matter what other players choose for their strategies, and better payoffs for at least one choice of strategies for the other players. BA 592 Lesson I.5 Simultaneous Move Theory 9

Example 3: Dominate Strategies The Prisoners’ dilemma described earlier is whether or not to confess. If you don’t confess, you get 1 year if the other prisoner does not confess and 15 years if he does. If you do confess, you get 0 if the other prisoner does not confess and 5 years if he does.

To decide whether to confess, define the normal form for the Prisoners’ Dilemma , and find any dominate strategies. BA 592 Lesson I.5 Simultaneous Move Theory 10

Example 3: Dominate Strategies Confess is a dominate strategy for each prisoner. In particular, it is the only Nash equilibrium.

Prisoner 1 Don't C.

Confess Prisoner 2 Don't C.

-1,-1 0,-15 Confess -15,0 -5,-5

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Example 3: Dominate Strategies The Price Competition Game modified by the 110% Low Price Guarantee has $95 as a weakly-dominate strategy for each player. Although it is not the only Nash equilibrium, it is the recommended solution.

Walmart $75 $85 $95 $75 0,0 0,0 0,0 Costco $85 0,0 500,500 500,500 $95 0,0 500,500 800,800

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Example 4: Iterated Dominance A dominated strategy for a player gives worse payoffs for that player compared with some other strategy, no matter what other players choose for their strategies. While dominate strategies are the recommended choice to play games, dominated strategies should never be chosen. Eliminating dominated strategies reduces the game, and the new game may have further dominated strategies, which can be eliminated, and so on.

A weakly dominated strategy for a player gives at least as bad payoffs for that player compared with some other strategy, no matter what other players choose for their strategies, and worse payoffs for at least one choice of strategies for the other players. Eliminating weakly-dominated strategies reduces the game, and the new game may have further weakly-dominated strategies, which can be eliminated, and so on.

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Example 4: Iterated Dominance The original Price Competition Game has $75 as a weakly dominated strategy for each player. Eliminate that strategy to solve the game.

Walmart $75 $85 $95 $75 0,0 0,0 0,0 Costco $85 0,0 500,500 0,1000 $95 0,0 1000,0 800,800

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Example 4: Iterated Dominance The reduced game has $85 as a dominate strategy for each player. It is thus the recommended solution to the original Price Competition Game after iterated elimination of weakly dominated strategies .

Costco Walmart $85 $95 $85 500,500 0,1000 $95 1000,0 800,800

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Example 5: Maximin Zero-sum games with two players can be solved using special analysis of the direct conflict of the players. Outcomes that are good for one player are bad for the other player. Hence, each player chooses his strategy by thinking: “Would this strategy be the best choice for me, given that my opponent choose the strategy that is worst for me?” A maximin strategy for a player is the action that yields the maximum payoff for that player given that the other player choses their strategy that gives the first player the minimum payoff.

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Example 5: Maximin Dallas Cowboys and New York Giants consider possible offensive (Cowboys) and defensive (Giants) plays as time runs out in the Super Bowl. Coaches for both teams have computed the probabilities percentages that the Cowboys win depending on which plays are selected.

Defense Offense Run Short Pass Med. Pass Long Pass Run 2 6 6 10 Pass 5 5.6

5 3 Blitz 13 10.5

1 0

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Example 5: Maximin Minimum (worst) outcomes for each offensive play are computed, and the offense chooses the maximin strategy Short Pass that maximizes those minimum payoffs.

Maximum (worst) outcomes for each defensive plays are computed, and the defense chooses the minimax strategy Pass that minimizes those maximum payoffs.

Defense Offense Run Pass Blitz Run Short Pass Med. Pass Long Pass 2 6 6 5 5.6

5 13 10.5

1 10 3 0 max = 10 max = 5.6 max = 13 min = 2 min = 5.6

min = 1 min = 0

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Example 5: Maximin Maximin and Minimax strategies are recommended in the football game because they are a Nash equilibrium: Short Pass for offense is a best response to Pass for defense, and Pass for defense is a best response to Short Pass for offense. (Because the game is zero sum, the game table need only list the payoffs for the row player.)

Defense Offense Run Pass Blitz Run Short Pass Med. Pass Long Pass 2 6 6 5 5.6

5 13 10.5

1 10 3 0 max = 10 max = 5.6 max = 13 min = 2 min = 5.6

min = 1 min = 0

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BA 592 Game Theory

End of Lesson I.5

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