Game Theory

Consider the Finger Game You want even, I will take odds. At the count of 3, we each hold out 1 or 2 fingers. If there are an even number of fingers, I pay you $1. If there are an odd number of fingers, you pay me $1.

Consider the Finger Game Payoff Table for me

Consider the Finger Game Payoff Table for me For each game there is a strategy For each game there is a payoff table for each player

Consider the Finger Game Payoff Table for me Assumptions Both players are rational Both players promote own welfare

Simple Games Two politicians are running for the U.S. Senate. Campaign plans must be made now for the final two days which each politician will spend in two key cities: Piedmont and New Underwood. They plan to spend one full day in each city or two full days in one city. Each campaign manager estimates the impact at the polls.

2 Person, 0 Sum Game Each player has one of 3 strategies Pay Off Table Spend one day in Piedmont, one day in New Underwood Spend two days in Piedmont Spend two days in New Underwood Pay Off Table

2 Person, 0 Sum Game 0 Sum because one wins, one loses Pay Off Table for Player (Politician) 1 If, player one spends one day in each city and player two spends two days in New Underwood, he would expect to win 4,000 votes, but if his opponent spends one day in each city, he would expect to gain only 1,000 votes.

2 Person, 0 Sum Game 0 Sum because one wins, one loses Pay Off Table for Player (Politician) 1 Pay off for player 2 is the negative of the payoff for player 1.

Dominated Strategy A strategy is dominated by a second strategy if the payoff is always as good as or better than the first. Strategy 3 is dominated by strategy 1

Dominated Strategy Player 2, believing player 1 is rational also knows player 1 will not consider strategy 3. For player 2, strategy 3 is dominated by both strategy 1 and strategy 2. Strategy 3 is eliminated.

Dominated Strategy For player 1, strategy 2 is now dominated by strategy 1 and is eliminated.

Dominated Strategy For player 1, strategy 2 is now dominated by strategy 1 and is eliminated.

Dominated Strategy For player 2, strategy 2 is now dominated by strategy 1 and is eliminated.

Dominated Strategy Both Players will now choose strategy 1.

No Dominate Strategy Suppose player 1 has the following payoff table.

No Dominate Strategy Strategy 1: Player 1 could win 6 but could lose as much as 3. Player 2, being rational will try to protect himself from a large payoff for player 1 so it seems likely player 1 would lose with strategy 1.

No Dominate Strategy Strategy 3: Player 1 could win 5 but could lose as much as 4. Player 2, being rational will again try to protect himself from a large payoff for player 1 so it seems likely player 1 would lose with strategy 3.

No Dominate Strategy Strategy 3: Player 1 could win 2 but will not lose anything regardless of player 2 strategy. Under strategy 2, player cannot lose and may win. Player 1 would likely choose strategy 2.

Player 2 Strategy Player 2 could lose as much as 5 or 6 with strategies 1 or 3 but will at least break even with strategy 2. Player 2 would likely choose strategy 2.

Summary Neither player improves her best chance of winning but both are forcing the opponent into the same position.

Summary By this reasoning, each player should minimize his maximum losses. Player 1 should select the minimum payoff to player 1. Player 2 should select the maximum payoff to player 1 is minimum.

Minimax, Maximin Criteria By this reasoning, each player should minimize his maximum losses. Player 1 should select the minimum payoff to player 1 is largest, maximin criteria. Player 2 should select the maximum payoff to player 1 is minimum, minimax criteria.

Maximin, Minimax Criteria Maximin and minimax result in same in the same payoff of 0 indicating this is a fair game. Payoff table has the same maximin and minimax criteria. This is because the minimum in the row and the maximum of its column results in the same entry. This is called a saddle point. maximin value minimax value

Maximin, Minimax Criteria Saddle point is crucial aspect of the game. If player 2 learns of player 1 strategy, she may elect to modify her strategy. Under this saddle point, no player has any motive to consider changing strategies. This is a stable or equilibrium solution. maximin value minimax value

Variation 3 Now consider a payoff where the maximin and minimax for each player is different. maximin value minimax value

Variation 3 Now consider a payoff where the maximin and minimax for each player is different. Player 1 can guarantee she will lose no more than 2 by strategy 1. Player 2 can guarantee she will lose no more than 2 by strategy 3. maximin value minimax value

Variation 3 Player 1 could win 2 from player 2 using maximin. Player 2 is smart and foresees this. Player 2 would rather select strategy 2 and win by 2. maximin value minimax value

Variation 3 Player 1 could win 2 from player 2 using maximin. Player 2 is smart and foresees this. Player 2 would rather select strategy 2 and win by 2. Player 1 would anticipate this switch and would switch to strategy 2 and win by 4. maximin value minimax value

Variation 3 Player 1 could win 2 from player 2 using maximin. Player 2 is smart and foresees this. Player 2 would rather select strategy 2 and win by 2. Player 1 would anticipate this switch and would switch to strategy 2 and win by 4. Player 2 anticipates this switch and switches back to strategy 3. maximin value minimax value

Variation 3 Player 1 could win 2 from player 2 using maximin. Player 2 is smart and foresees this. Player 2 would rather select strategy 2 and win by 2. Player 1 would anticipate this switch and would switch to strategy 2 and win by 4. Player 2 anticipates this switch and switches back to strategy 3. Possibility of this switch causes player 1 to again switch to strategy 1. maximin value minimax value

Variation 3 Cycle repeats; this is an unstable solution. maximin value Player 1 could win 2 from player 2 using maximin. Player 2 is smart and foresees this. Player 2 would rather select strategy 2 and win by 2. Player 1 would anticipate this switch and would switch to strategy 2 and win by 4. Player 2 anticipates this switch and switches back to strategy 3. Possibility of this switch causes player 1 to again switch to strategy 1. Cycle repeats; this is an unstable solution. maximin value minimax value

Key Points Whenever one player’s strategy is predictable, the opponent can take advantage of the information to improve his or her position. Essential feature of a rational plan for playing a game is that neither player should be able to deduce the strategy to be used by the other. Choice of strategy under this circumstance is random choice. Ex: Finger odds or even game.

Games with Mixed Strategies If there is no saddle point, game theory suggests each player to assign a probability distribution over the strategy set. Let xi = probability that player 1 will use strategy i yj = probability that player 2 will use strategy j pij = payoff if player 1 uses strategy i and player 2 uses strategy j

Games with Mixed Strategies

Games with Mixed Strategies

Minimax Theorem If mixed strategies are allowed, the pair of mixed strategies that is optimal according to the minimax criteria provides a stable solution with , so that neither player can do better by unilaterally changing his or her strategy. Minimax says that a player should select a mixed strategy that minimizes the expected loss to him or herself.

Graphical Solution Consider a reduced payoff for player 1 Expected payoff for player 1 then is

Graphical Solution 5-5x1 4-6x1 -3+5x1 Maximin Point

Graphical Solution 5-5x1 4-6x1 -3+5x1 Maximin Point

Algebraic Solution Player 2 will never choose top line because he/she wants to minimize payoff to player 1. Player 1 wants to maximize minimum payoff, then

Algebraic Solution If player is operating optimally then,

Algebraic Solution We also know that the optimal solution for player 1 is x* = (7/11, 4/11). Therefore,

Algebraic Solution Now if y1*>0, we are on the blue line and Above the maximin pt. Therefore, y1*=0.

Algebraic Solution Therefore, Select two values for x1, say 0 and 1 to solve for y2* and y3*

Algebraic Solution At x1 = 0 At x1 = 1

Algebraic Solution At x1 = 0 At x1 = 1 Equating gives

LP Formulation Recall, and, by maximin theorem, Further, these must hold for each strategy of player 2 including the pure stategies; e.g., y = (1, 0, 0), y = (0, 1, 0), y = (0, 0, 1)

LP Formulation Then, for each pure strategy, If we plug this back into the expected payoff equation, we get

LP Formulation Since,

LP Formulation We now have

LP Formulation The problem is, we don’t have an objective function or knowledge of what is. We solve this by adding an artificial variable, for and maximizing it which, by minimax theorem, will equal

LP Formulation Similarly, player 2 would like to minimize the maximum payoff to player 1. Then, by the same logic,

Political Campaign Problem Let’s reconsider our revised political campaign problem.

Political Campaign Problem For player 2,