Finite Mathematics & Its Applications, 10/e by Goldstein/Schneider/SiegelCopyright © 2010 Pearson Education, Inc. 1 of 68 Chapter 9 The Theory of Games

Finite Mathematics & Its Applications, 10/e by Goldstein/Schneider/SiegelCopyright © 2010 Pearson Education, Inc. 2 of 68 Outline 9.1 Games and Strategies 9.2 Mixed Strategies 9.3 Determining Optimal Mixed Strategies

Finite Mathematics & Its Applications, 10/e by Goldstein/Schneider/SiegelCopyright © 2010 Pearson Education, Inc. 3 of Games and Strategies 1.Fundamental Problem of Game Theory 2.Payoff Matrix 3.Strategies 4.Zero-Sum Game 5.Optimal Pure Strategy for R 6.Optimal Pure Strategy for C 7.Saddle Point 8.Strictly Determined Game and Its Value

Finite Mathematics & Its Applications, 10/e by Goldstein/Schneider/SiegelCopyright © 2010 Pearson Education, Inc. 4 of 68 Fundamental Problem of Game Theory A mathematical game consists of two or more players. Each player is allowed to make a move. As the result of a move by each player, there is a payoff to each player. Fundamental Problem of Game Theory: How should each player decide his move in order to maximize his gain?

Finite Mathematics & Its Applications, 10/e by Goldstein/Schneider/SiegelCopyright © 2010 Pearson Education, Inc. 5 of 68 Two furniture stores, Reliable and Cut-Rate, in a town are planning a Labor Day sale and each has the option of marking its furniture down by 10% or 20%. If both choose a 10% discount, then Reliable will capture 60% of the sales. But if Reliable chooses 10% and Cut-Rate 20%, then Reliable will capture 35% of the sales. If Reliable chooses 20% and Cut-Rate 10%, Reliable will capture 80% of the sales. If they both choose 20%, Reliable will capture 50% of the sales. What percentage discount should each store choose? Example Mathematical Game

Finite Mathematics & Its Applications, 10/e by Goldstein/Schneider/SiegelCopyright © 2010 Pearson Education, Inc. 6 of 68 Payoff Matrix The payoff matrix shown above is an mxn matrix. The rows correspond to a move by R and the entries represent the gain to R depending upon C's move. The columns correspond to a move by C.

Finite Mathematics & Its Applications, 10/e by Goldstein/Schneider/SiegelCopyright © 2010 Pearson Education, Inc. 7 of 68 Example Payoff Matrix Write the payoff matrix for the furniture store problem in the first example. Let Reliable be R and Cut-Rate be C. The moves are "10% discount" or "20% discount". If R = C = 10%, R's gain is.6 (60% of the sales). If R = 10% and C = 20%, R's gain is.35. If R = 20% and C = 10%, R's gain is.8. If R = C = 20%, R's gain is.5.

Finite Mathematics & Its Applications, 10/e by Goldstein/Schneider/SiegelCopyright © 2010 Pearson Education, Inc. 8 of 68 Example Payoff Matrix (2) Collecting this information in a matrix gives the payoff matrix:.

Finite Mathematics & Its Applications, 10/e by Goldstein/Schneider/SiegelCopyright © 2010 Pearson Education, Inc. 9 of 68 Strategies If a game is played repeatedly, the players can adopt various strategies to attempt to maximize their respective gains (or minimize their losses). A pure strategy is one in which a player, on consecutive plays, consistently chooses the same row (or column). Strategies involving varied moves are called mixed strategies.

Finite Mathematics & Its Applications, 10/e by Goldstein/Schneider/SiegelCopyright © 2010 Pearson Education, Inc. 10 of 68 For the furniture store problem from the first example, find the pure strategy that will maximize Reliable's share of sales and minimize Cut-Rate's loss of sales. Example Strategies

Finite Mathematics & Its Applications, 10/e by Goldstein/Schneider/SiegelCopyright © 2010 Pearson Education, Inc. 11 of 68 If Reliable chooses the 10% discount, then Cut- Rate will choose 20%. This will result in Reliable getting 35% of the sales. If Reliable chooses the 20% discount, then Cut- Rate will choose 20%. This will result in Reliable getting 50% of the sales. Therefore, the pure optimal strategy is for both Reliable and Cut-Rate to choose 20% discounts. Example Strategies

Finite Mathematics & Its Applications, 10/e by Goldstein/Schneider/SiegelCopyright © 2010 Pearson Education, Inc. 12 of 68 Zero-Sum Game A game in which a payoff for R (row player) of a given amount results in a loss to C (column player) of the same amount, and vice versa, is called a zero-sum game.

Finite Mathematics & Its Applications, 10/e by Goldstein/Schneider/SiegelCopyright © 2010 Pearson Education, Inc. 13 of 68 Suppose that R and C play a coin-matching game. If R and C both show heads, then C pays R $5. If R shows heads and C shows tails, the R pays C $8. If R shows tails and C shows heads, then C pays R $3. If both show tails, then C pays R $1. a) Determine the payoff matrix b) Suppose R and C play the game repeatedly. Determine optimal pure strategies for R and C. Example Zero-Sum Game

Finite Mathematics & Its Applications, 10/e by Goldstein/Schneider/SiegelCopyright © 2010 Pearson Education, Inc. 14 of 68 This is a zero-sum game. In the payoff matrix for R, each entry specifies a payoff from C to R and a negative entry denotes a loss to R (that is, a payoff to C). The payoff matrix is Example Zero-Sum Game (a).

Finite Mathematics & Its Applications, 10/e by Goldstein/Schneider/SiegelCopyright © 2010 Pearson Education, Inc. 15 of 68 If R consistently shows heads so as to maybe gain $5, C would show tails causing R to lose $8. If R consistently shows tails, then C would also show tails causing R to only win $1 instead of $3. Therefore, the optimal pure strategy is for both R and C to show tails. Example Zero-Sum Game (b)

Finite Mathematics & Its Applications, 10/e by Goldstein/Schneider/SiegelCopyright © 2010 Pearson Education, Inc. 16 of 68 Optimal Pure Strategy for R 1. For each row of the payoff matrix, determine the least element. 2. Choose the row for which this element is as large as possible.

Finite Mathematics & Its Applications, 10/e by Goldstein/Schneider/SiegelCopyright © 2010 Pearson Education, Inc. 17 of 68 Optimal Pure Strategy for C 1. For each column of the payoff matrix, determine the largest element. 2. Choose the column for which this element is a small as possible.

Finite Mathematics & Its Applications, 10/e by Goldstein/Schneider/SiegelCopyright © 2010 Pearson Education, Inc. 18 of 68 Example Optimal Pure Strategy Determine optimal pure strategies for R and C for the game whose payoff matrix is

Finite Mathematics & Its Applications, 10/e by Goldstein/Schneider/SiegelCopyright © 2010 Pearson Education, Inc. 19 of 68 Example Optimal Pure Strategy for R 1. Circle least element in each row. 2. Choose the row with the largest circled entry. Therefore, R should choose row 2.

Finite Mathematics & Its Applications, 10/e by Goldstein/Schneider/SiegelCopyright © 2010 Pearson Education, Inc. 20 of 68 Example Optimal Pure Strategy for C 1. Circle largest element in each column. 2. Choose the column with the smallest circled entry. Therefore, C should choose column 1.

Finite Mathematics & Its Applications, 10/e by Goldstein/Schneider/SiegelCopyright © 2010 Pearson Education, Inc. 21 of 68 Saddle Point In the previous payoff matrix, there is an entry that is simultaneously the minimum element in its row and the maximum element in its column. Such an entry is called a saddle point for the game. A game can have no saddle points or more than one saddle point. If it has more than one, then the saddle points are equal.

Finite Mathematics & Its Applications, 10/e by Goldstein/Schneider/SiegelCopyright © 2010 Pearson Education, Inc. 22 of 68 Strictly Determined Game and Its Value A game that has a saddle point is called a strictly determined game. The optimal pure strategy for R is to choose the row containing the saddle point and for C to choose the column containing the saddle point. The saddle point in a strictly determined game is also called the value of the game and is the payoff to R if both players use their pure optimal strategies.

Finite Mathematics & Its Applications, 10/e by Goldstein/Schneider/SiegelCopyright © 2010 Pearson Education, Inc. 23 of 68 Example Saddle Point Find the saddle point and value of the game with payoff matrix The saddle point is the 1 in the second row and first column. The value of the game is 1.

Finite Mathematics & Its Applications, 10/e by Goldstein/Schneider/SiegelCopyright © 2010 Pearson Education, Inc. 24 of 68  A zero-sum game is one in which a payoff to one player results in a loss of the same amount to the other player.  The entry in the i th row and j th column of a payoff matrix gives the payoff to the row player (equivalently, the loss to the column player) when the row player chooses row i and the column player chooses column j. Summary Section Part 1

Finite Mathematics & Its Applications, 10/e by Goldstein/Schneider/SiegelCopyright © 2010 Pearson Education, Inc. 25 of 68  A pure strategy is one in which the player consistently chooses the same row or column. Strategies involving varied moves are called mixed strategies.  The optimal pure strategy for the row player is to choose the row whose least element is maximal. The optimal pure strategy for the column player is to choose the column whose greatest element is minimal. Summary Section Part 2

Finite Mathematics & Its Applications, 10/e by Goldstein/Schneider/SiegelCopyright © 2010 Pearson Education, Inc. 26 of 68  A saddle point is an entry in a payoff matrix that is simultaneously the least element of its row and the greatest element of its column. A game need not have a saddle point. If a game has more than one saddle point, then the saddle points are equal.  A game with a saddle point is called a strictly determined game. For such a game, the optimal pure strategy for each player is to choose a row or column containing a saddle point. Summary Section Part 3

Finite Mathematics & Its Applications, 10/e by Goldstein/Schneider/SiegelCopyright © 2010 Pearson Education, Inc. 27 of 68  In a strictly determined game, if both players use optimal pure strategies, then the saddle point gives the payoff to the row player. The value of the saddle point is called the value of the game. Summary Section Part 4

Finite Mathematics & Its Applications, 10/e by Goldstein/Schneider/SiegelCopyright © 2010 Pearson Education, Inc. 28 of Mixed Strategies 1.Game With No Saddle Point 2.Mixed Strategy 3.Expected Value of a Pair of Strategies

Finite Mathematics & Its Applications, 10/e by Goldstein/Schneider/SiegelCopyright © 2010 Pearson Education, Inc. 29 of 68 Example Game With No Saddle Point A game has the payoff matrix a) Show that there is no saddle point. b) Analyze what would happen if R or C consistently played their optimal pure strategy.

Finite Mathematics & Its Applications, 10/e by Goldstein/Schneider/SiegelCopyright © 2010 Pearson Education, Inc. 30 of 68 Find the largest minimum element in each row. Find the smallest maximum element in each column. There is no saddle point because the entry circled for the row choice was -1 and for the column choice was 2. Example Game With No Saddle Point (a)

Finite Mathematics & Its Applications, 10/e by Goldstein/Schneider/SiegelCopyright © 2010 Pearson Education, Inc. 31 of 68 Example Game With No Saddle Point (b) If R consistently plays row 1 and C column 1, R will eventually wise up and start playing row 2 to gain 2 instead of lose 1. If R then consistently plays row 2, C will eventually wise up and start playing column 2 to gain 3. This will cause R to eventually play row 1. No pure strategy both maximizes R's gain and minimizes C's loss.

Finite Mathematics & Its Applications, 10/e by Goldstein/Schneider/SiegelCopyright © 2010 Pearson Education, Inc. 32 of 68 Mixed Strategy A mixed strategy for R is a row matrix that contains the probabilities that R will choose that row to play. A mixed strategy for C is a column matrix that contains the probabilities that C will choose that column to play.

Finite Mathematics & Its Applications, 10/e by Goldstein/Schneider/SiegelCopyright © 2010 Pearson Education, Inc. 33 of 68 Expected Value of a Pair of Strategies Expected Value of a Pair of Strategies If a game has payoff matrix R plays the strategy

Finite Mathematics & Its Applications, 10/e by Goldstein/Schneider/SiegelCopyright © 2010 Pearson Education, Inc. 34 of 68 Expected Value of a Pair of Strategies (2) and C plays then the expected value, e, of the pair of strategies is

Finite Mathematics & Its Applications, 10/e by Goldstein/Schneider/SiegelCopyright © 2010 Pearson Education, Inc. 35 of 68 Example Mixed Strategy For the game with playoff matrix assume R will play each row with a probability of.5. Which of the following strategies is more advantageous for C?

Finite Mathematics & Its Applications, 10/e by Goldstein/Schneider/SiegelCopyright © 2010 Pearson Education, Inc. 36 of 68 Example Mixed Strategy (2) C should use strategy A with an expected gain to R of.75 over strategy B with an expected gain to R of.8.

Finite Mathematics & Its Applications, 10/e by Goldstein/Schneider/SiegelCopyright © 2010 Pearson Education, Inc. 37 of 68  If a game is not strictly determined, then the players should use mixed strategies. A mixed strategy for the row player is a row matrix whose i th entry is the probability that the row player will choose row i on any repetition of the game. A mixed strategy for the column player is a column matrix whose j th entry is the probability that the column player will choose column j on any repetition of the game. Summary Section Part 1

Finite Mathematics & Its Applications, 10/e by Goldstein/Schneider/SiegelCopyright © 2010 Pearson Education, Inc. 38 of 68  The expected value of a pair of mixed strategies R = [r 1 r 2 … r m ] and C = [c 1 c 2 … c n ] T is the average payoff per game to the row player if these mixed strategies are used. The expected value of the pair R, C of mixed strategies is found by computing the product RAC, where A is the payoff matrix for the game. Summary Section Part 2

Finite Mathematics & Its Applications, 10/e by Goldstein/Schneider/SiegelCopyright © 2010 Pearson Education, Inc. 39 of Determining Optimal Mixed Strategies 1.Adding Constant to Payoff Matrix 2.Optimal Strategy for R 3.Optimal Strategy for C 4.Determining Optimal Strategy for R 5.Determining Optimal Strategy for C 6.Dual Problem 7.Fundamental Theorem of Game Theory

Finite Mathematics & Its Applications, 10/e by Goldstein/Schneider/SiegelCopyright © 2010 Pearson Education, Inc. 40 of 68 Adding Constant to Payoff Matrix Let A be a payoff matrix for a game with expected value e. Let B be a payoff matrix obtained from A by adding a fixed constant k to every entry of A. Then the optimal strategies for the players are the same for A and B. Also the expected value of B = k + e.

Finite Mathematics & Its Applications, 10/e by Goldstein/Schneider/SiegelCopyright © 2010 Pearson Education, Inc. 41 of 68 Optimal Strategy for R Optimal Strategy for RTo every choice of a strategy for R there is a best counterstrategy - that is, a strategy for C that results in the least expected value e. An optimal mixed strategy for R is one for which the expected value against C's best counterstrategy is as large as possible.

Finite Mathematics & Its Applications, 10/e by Goldstein/Schneider/SiegelCopyright © 2010 Pearson Education, Inc. 42 of 68 Optimal Strategy for C Optimal Strategy for CTo every choice of a strategy for C there is a best counterstrategy - that is, a strategy for R that results in the largest expected value e. An optimal mixed strategy for C is one for which the expected value against R's best counterstrategy is as small as possible.

Finite Mathematics & Its Applications, 10/e by Goldstein/Schneider/SiegelCopyright © 2010 Pearson Education, Inc. 43 of 68 Determining Optimal Strategy for R Optimal Strategy for RLet the payoff matrix of a game be where all entries of the matrix are positive numbers.

Finite Mathematics & Its Applications, 10/e by Goldstein/Schneider/SiegelCopyright © 2010 Pearson Education, Inc. 44 of 68 Determining Optimal Strategy for R (2) Let y 1, y 2,…,y m be chosen so as to minimize y 1 + y 2 + … + y m subject to the constraints

Finite Mathematics & Its Applications, 10/e by Goldstein/Schneider/SiegelCopyright © 2010 Pearson Education, Inc. 45 of 68 Determining Optimal Strategy for R (3) Let Then an optimal strategy for R is [r 1 r 2 … r m ], where r 1 = vy 1, r 2 = vy 2, …, r m = vy m. Furthermore, if C adopts the best counterstrategy, then the expected value is v.

Finite Mathematics & Its Applications, 10/e by Goldstein/Schneider/SiegelCopyright © 2010 Pearson Education, Inc. 46 of 68 Example Optimal Strategy for R Let a game have payoff matrix a) Determine an optimal strategy for R. b) Determine the expected payoff to R if C uses the best counterstrategy.

Finite Mathematics & Its Applications, 10/e by Goldstein/Schneider/SiegelCopyright © 2010 Pearson Education, Inc. 47 of 68 Example Optimal Strategy for R (2) Add a constant to every entry of so that the new matrix has all positive entries. To make all entries positive, we can add 4 to each entry.

Finite Mathematics & Its Applications, 10/e by Goldstein/Schneider/SiegelCopyright © 2010 Pearson Education, Inc. 48 of 68 Example Optimal Strategy for R (3) We need to solve minimize y 1 + y 2 subject to the constraints

Finite Mathematics & Its Applications, 10/e by Goldstein/Schneider/SiegelCopyright © 2010 Pearson Education, Inc. 49 of 68 Cornery 1 + y 2 (0,1)1 (5/51,2/17)11/51 (1/3,0)1/3 Example Optimal Strategy for R (4) The sketch of the feasible set is

Finite Mathematics & Its Applications, 10/e by Goldstein/Schneider/SiegelCopyright © 2010 Pearson Education, Inc. 50 of 68 Example Optimal Strategy for R (5) Solution is Let The optimal strategy for R is.

Finite Mathematics & Its Applications, 10/e by Goldstein/Schneider/SiegelCopyright © 2010 Pearson Education, Inc. 51 of 68 Example Optimal Strategy for R (6) If C adopts the best counterstrategy, then the expected value is Note: Remember, we added 4 to all entries of the payoff matrix so the expected value of the original problem is v - 4.

Finite Mathematics & Its Applications, 10/e by Goldstein/Schneider/SiegelCopyright © 2010 Pearson Education, Inc. 52 of 68 Determining Optimal Strategy for C Optimal Strategy for CLet the payoff matrix of a game be where all entries of the matrix are positive numbers.

Finite Mathematics & Its Applications, 10/e by Goldstein/Schneider/SiegelCopyright © 2010 Pearson Education, Inc. 53 of 68 Determining Optimal Strategy for C (2) Let z 1, z 2,…,z n be chosen so as to maximize z 1 + z 2 + … + z n subject to the constraints

Finite Mathematics & Its Applications, 10/e by Goldstein/Schneider/SiegelCopyright © 2010 Pearson Education, Inc. 54 of 68 Determining Optimal Strategy for C (3) Let Then an optimal strategy for C is [c 1 c 2 … c n ] T, where c 1 = vz 1, c 2 = vz 2, …, c n = vz n. Furthermore, if R adopts the best counterstrategy, then the expected value is v.

Finite Mathematics & Its Applications, 10/e by Goldstein/Schneider/SiegelCopyright © 2010 Pearson Education, Inc. 55 of 68 Example Optimal Strategy for C Let a game have payoff matrix a) Determine an optimal strategy for C. b) Determine the expected payoff to R if R uses the best counterstrategy.

Finite Mathematics & Its Applications, 10/e by Goldstein/Schneider/SiegelCopyright © 2010 Pearson Education, Inc. 56 of 68 Example Optimal Strategy for C (2) Adding 4 to every entry of results in

Finite Mathematics & Its Applications, 10/e by Goldstein/Schneider/SiegelCopyright © 2010 Pearson Education, Inc. 57 of 68 Example Optimal Strategy for C (3) We need to solve maximize z 1 + z 2 subject to the constraints

Finite Mathematics & Its Applications, 10/e by Goldstein/Schneider/SiegelCopyright © 2010 Pearson Education, Inc. 58 of 68 Cornerz 1 + z 2 (0,0)0 (0,1/9)1/9 (8/51,1/17)11/51 (1/6,0)1/6 Example Optimal Strategy for C (4) The sketch of the feasible set is

Finite Mathematics & Its Applications, 10/e by Goldstein/Schneider/SiegelCopyright © 2010 Pearson Education, Inc. 59 of 68 Example Optimal Strategy for C (5) Solution is Let The optimal strategy for C is

Finite Mathematics & Its Applications, 10/e by Goldstein/Schneider/SiegelCopyright © 2010 Pearson Education, Inc. 60 of 68 Example Optimal Strategy for C (6) If R adopts the best counterstrategy, then the expected payoff to R is Note: This is the same value we got when solving for the optimal strategy for R.

Finite Mathematics & Its Applications, 10/e by Goldstein/Schneider/SiegelCopyright © 2010 Pearson Education, Inc. 61 of 68 Dual Problem The problem of finding the optimal strategy for R is a linear programming problem whose dual is the problem of finding an optimal strategy for C, and vice versa.

Finite Mathematics & Its Applications, 10/e by Goldstein/Schneider/SiegelCopyright © 2010 Pearson Education, Inc. 62 of 68 Example Dual Problem Use the simplex method to determine the optimal strategies for the game with payoff matrix

Finite Mathematics & Its Applications, 10/e by Goldstein/Schneider/SiegelCopyright © 2010 Pearson Education, Inc. 63 of 68 Example Dual Problem (2) The linear programming problem is Introducing slack variables t and u, the initial and final tableaux are:

Finite Mathematics & Its Applications, 10/e by Goldstein/Schneider/SiegelCopyright © 2010 Pearson Education, Inc. 64 of 68 Example Dual Problem (3)

Finite Mathematics & Its Applications, 10/e by Goldstein/Schneider/SiegelCopyright © 2010 Pearson Education, Inc. 65 of 68 Example Dual Problem (4) The solution is z 1 = 1/17, z 2 = 4/17 and M = 5/17, so v = 17/5. The optimal strategy for C is The solution to the dual problem is y 1 = t = 3/17, y 2 = u = 2/17. The optimal strategy for R is

Finite Mathematics & Its Applications, 10/e by Goldstein/Schneider/SiegelCopyright © 2010 Pearson Education, Inc. 66 of 68 Fundamental Theorem of Game Theory Every two-person zero-sum game has a solution.

Finite Mathematics & Its Applications, 10/e by Goldstein/Schneider/SiegelCopyright © 2010 Pearson Education, Inc. 67 of 68 Summary Section Part 1  An optimal mixed strategy for the row player is one for which the column player's best counterstrategy results in the greatest possible expected value. Similarly, an optimal mixed strategy for the column player is one for which the row player's best counterstrategy results in the least possible expected value.

Finite Mathematics & Its Applications, 10/e by Goldstein/Schneider/SiegelCopyright © 2010 Pearson Education, Inc. 68 of 68 Summary Section Part 2  Optimal mixed strategies for each player can be found by solving a pair of dual linear programming problems.