Game Theory Objectives Understand the principles of zerosum
Game Theory Objectives: • Understand the principles of zero-sum, twoplayer game • Analyzing pure strategy game, dominance principle. • Solve mixed strategy games. Quantitative Methods for Business Slide 1
Introduction to Game Theory n n In decision analysis, a single decision maker seeks to select an optimal alternative. In game theory, there are two or more decision makers, called players, who compete as adversaries against each other. It is assumed that each player has the same information and will select the strategy that provides the best possible outcome from his point of view. Each player selects a strategy independently without knowing in advance the strategy of the other player(s). continue Quantitative Methods for Business Slide 2
Introduction to Game Theory n n n The combination of the competing strategies provides the value of the game to the players. Game models classification according to: number of players, sum of all payoffs, number of strategies employed Examples of competing players are teams, armies, companies, political candidates, and contract bidders. Quantitative Methods for Business Slide 3
Two-Person Zero-Sum Game n n Two-person means there are two competing players in the game. Zero-sum means the gain (or loss) for one player is equal to the corresponding loss (or gain) for the other player. The gain and loss balance out so that there is a zero-sum for the game. What one player wins, the other player loses. Quantitative Methods for Business Slide 4
Two-Person Zero-Sum Game Example n Competing for Vehicle Sales Suppose that there are only two vehicle dealer-ships in a small city. Each dealership is considering three strategies that are designed to take sales of new vehicles from the other dealership over a four-month period. The strategies, assumed to be the same for both dealerships. Quantitative Methods for Business Slide 5
Two-Person Zero-Sum Game Example n Strategy Choices Strategy 1: Offer a cash rebate on a new vehicle. Strategy 2: Offer free optional equipment on a new vehicle. Strategy 3: Offer a 0% loan on a new vehicle. Quantitative Methods for Business Slide 6
Two-Person Zero-Sum Game Example n Payoff Table: Number of Vehicle Sales Gained Per Week by Dealership A (or Lost Per Week by Dealership B) Dealership B Dealership A Cash Rebate b 1 Free Options b 2 0% Loan b 3 Cash Rebate a 1 Free Options a 2 0% Loan a 3 2 -3 3 2 3 -2 1 -1 0 Quantitative Methods for Business Slide 7
Two-Person Zero-Sum Game Example n Step 1: Identify the minimum payoff for each row (for Player A). n Step 2: For Player A, select the strategy that provides the maximum of the row minimums (called the maximin). Quantitative Methods for Business Slide 8
Two-Person Zero-Sum Game Example n Identifying Maximin and Best Strategy Dealership B Dealership A Cash Rebate b 1 Free Options b 2 0% Loan b 3 Row Minimum Cash Rebate a 1 Free Options a 2 0% Loan a 3 2 -3 3 2 3 -2 1 -1 0 1 -3 -2 Best Strategy For Player A Quantitative Methods for Business Maximin Payoff Slide 9
Two-Person Zero-Sum Game Example n n Step 3: Identify the maximum payoff for each column (for Player B). Step 4: For Player B, select the strategy that provides the minimum of the column maximums (called the minimax). Quantitative Methods for Business Slide 10
Two-Person Zero-Sum Game Example n Identifying Minimax and Best Strategy Dealership B Dealership A Cash Rebate b 1 Free Options b 2 0% Loan b 3 Cash Rebate a 1 Free Options a 2 0% Loan a 3 2 -3 3 2 3 -2 1 -1 0 Column Maximum 3 3 1 Quantitative Methods for Business Best Strategy For Player B Minimax Payoff Slide 11
Pure Strategy n Whenever an optimal pure strategy exists: n the maximum of the row minimums equals the minimum of the column maximums (Player A’s maximin equals Player B’s minimax) n the game is said to have a saddle point (the intersection of the optimal strategies) n the value of the saddle point is the value of the game n neither player can improve his/her outcome by changing strategies even if he/she learns in advance the opponent’s strategy Quantitative Methods for Business Slide 12
Pure Strategy Example n Saddle Point and Value of the Game Dealership B Value of the game is 1 Dealership A Cash Rebate b 1 Free Options b 2 0% Loan b 3 Cash Rebate a 1 Free Options a 2 0% Loan a 3 2 -3 3 2 3 -2 1 -1 0 1 -3 -2 Column Maximum 3 3 1 Saddle Point Quantitative Methods for Business Row Minimum Slide 13
Pure Strategy Example n Pure Strategy Summary n Player A should choose Strategy a 1 (offer a cash rebate). n Player A can expect a gain of at least 1 vehicle sale per week. n Player B should choose Strategy b 3 (offer a 0% loan). n Player B can expect a loss of no more than 1 vehicle sale per week. Quantitative Methods for Business Slide 14
Mixed Strategy n n n If the maximin value for Player A does not equal the minimax value for Player B, then a pure strategy is not optimal for the game. In this case, a mixed strategy is best. With a mixed strategy, each player employs more than one strategy. Each player should use one strategy some of the time and other strategies the rest of the time. The optimal solution is the relative frequencies with which each player should use his possible strategies. Quantitative Methods for Business Slide 15
Mixed Strategy Example n Consider the following two-person zero-sum game. The maximin does not equal the minimax. There is not an optimal pure strategy. Player B Player A b 1 b 2 a 1 a 2 4 11 8 5 Column Maximum 11 8 Quantitative Methods for Business Row Minimum 4 Maximin 5 Minimax Slide 16
Mixed Strategy Example p = the probability Player A selects strategy a 1 (1 - p) = the probability Player A selects strategy a 2 If Player B selects b 1: EV = 4 p + 11(1 – p) If Player B selects b 2: EV = 8 p + 5(1 – p) Quantitative Methods for Business Slide 17
Mixed Strategy Example To solve for the optimal probabilities for Player A we set the two expected values equal and solve for the value of p. 4 p + 11(1 – p) = 8 p + 5(1 – p) 4 p + 11 – 11 p = 8 p + 5 – 5 p 11 – 7 p = 5 + 3 p Hence, -10 p = -6 (1 - p) =. 4 p =. 6 Player A should select: Strategy a 1 with a. 6 probability and Strategy a 2 with a. 4 probability. Quantitative Methods for Business Slide 18
Mixed Strategy Example q = the probability Player B selects strategy b 1 (1 - q) = the probability Player B selects strategy b 2 If Player A selects a 1: EV = 4 q + 8(1 – q) If Player A selects a 2: EV = 11 q + 5(1 – q) Quantitative Methods for Business Slide 19
Mixed Strategy Example To solve for the optimal probabilities for Player B we set the two expected values equal and solve for the value of q. 4 q + 8(1 – q) = 11 q + 5(1 – q) 4 q + 8 – 8 q = 11 q + 5 – 5 q 8 – 4 q = 5 + 6 q Hence, -10 q = -3 (1 - q) =. 7 q =. 3 Player B should select: Strategy b 1 with a. 3 probability and Strategy b 2 with a. 7 probability. Quantitative Methods for Business Slide 20
Mixed Strategy Example n Value of the Game For Player A: EV = 4 p + 11(1 – p) = 4(. 6) + 11(. 4) = 6. 8 For Player B: EV = 4 q + 8(1 – q) = 4(. 3) + 8(. 7) = 6. 8 Quantitative Methods for Business Expected gain per game for Player A Expected loss per game for Player B Slide 21
Dominated Strategies Example Suppose that the payoff table for a two-person zerosum game is the following. Here there is no optimal pure strategy. Player B Player A b 1 b 2 b 3 a 1 a 2 a 3 6 1 3 5 0 4 -2 3 -3 Column Maximum 6 5 3 Quantitative Methods for Business Row Minimum -2 0 -3 Maximin Minimax Slide 22
Dominated Strategies Example If a game larger than 2 x 2 has a mixed strategy, we first look for dominated strategies in order to reduce the size of the game. Player B Player A b 1 b 2 b 3 a 1 a 2 a 3 6 1 3 5 0 4 -2 3 -3 Player A’s Strategy a 3 is dominated by Strategy a 1, so Strategy a 3 can be eliminated. Quantitative Methods for Business Slide 23
Dominated Strategies Example We continue to look for dominated strategies in order to reduce the size of the game. Player B Player A b 1 b 2 b 3 a 1 a 2 6 1 5 0 -2 3 Player B’s Strategy b 1 is dominated by Strategy b 2, so Strategy b 1 can be eliminated. Quantitative Methods for Business Slide 24
Dominated Strategies Example The 3 x 3 game has been reduced to a 2 x 2. It is now possible to solve algebraically for the optimal mixed -strategy probabilities. Player B Player A b 2 b 3 a 1 a 2 5 0 -2 3 For Player A: p = 0. 3; For Player B: EV = 1. 5 q = 0. 5; Quantitative Methods for Business EV = 1. 5 Slide 25
Prisoner’s Dilemma Two thieves were caught because they stole a large amount of money in a bank. However, the police did not have enough evidence to convict them. They could send both two of them to prison for two years with the evidence using the stolen motorcycle. They decided to separate thieves into different prison room. And they suggested to each thief: B Confess Not Confess Both get 3 years 0 year for A 4 years for B Not Confess 4 years for A 0 year for B Both get 2 years A Quantitative Methods for Business Slide 26
Other Game Theory Models n n n Two-Person, Constant-Sum Games (The sum of the payoffs is a constant other than zero. ) Variable-Sum Games (The sum of the payoffs is variable. ) n-Person Games (A game involves more than two players. ) Cooperative Games (Players are allowed pre-play communications. ) Infinite-Strategies Games (An infinite number of strategies are available for the players. ) Quantitative Methods for Business Slide 27
Homework 02 n Module 4: M 4. 8, M 4. 11, M 4. 12, M 4. 13, M 4. 14, M 4. 18 Quantitative Methods for Business Slide 31
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