Discrete Math Matrix Applications 7 4 Game Theory

Discrete Math Matrix Applications 7. 4 Game Theory: Introduction

Essential Questions 1. How do you use matrices to organize, manipulate, and display information? 2. How can matrices be used to model real world phenomenon? 12/16/2021 2

Students shall be able to Describe and analyze multiplayer games in terms of strategies with matrices. Determine whether a game is strictly determined. Find a game’s saddle-point of a game. 12/16/2021 3

Agenda 12/16/2021 Warm up Homework review Lesson / activity Class work Summary 4

Communicate Effectively Game Theory Assumptions: (For this course) ◦ Each player plays to win. ◦ The rules are knowable and logically coherent so that a Payoff matrix can be developed. 12/16/2021 Strategy: planned alternative course of action (aka strategic plan). Strictly determined: Every player has the same strategy regardless of what the other does. 5

Communicate Effectively 12/16/2021 Payoff Matrix: A player oriented matrix that shows the payoff for different the choices: Dominated Strategy: A strategy which will always loses to another player’s strategy. 6

Tina & Sol’s Coin Match Game (2 player game) Tina has a game she wants to play with Sol. Rules: ◦ Each player hides and then at the same time shows their choice. Each player has two sided coin. ◦ When both players show heads, Sol wins 3 coins. ◦ When both players show tails then Tina wins 2 coins. ◦ When the coins do not match then Tina wins 1 coin. 12/16/2021 7

Communicate Effectively Payoff Matrix: A player oriented matrix that shows the payoff: ◦ (First from Sol then from Tina’s perspective: ) 12/16/2021 8

Communicate Effectively 12/16/2021 Row Minimum: Best strategy the row player in a strictly determined game is to select the largest value. Maximin: the maximum of the row minimums. Column Maximum: Best strategy for the column player is associated with the smallest value. Minimax: the minimum of the column maximums. Strictly Determined: A game whose maximin and minimax values are the same. This value, Saddle Point is the amount won per play. 9

Finding a strategy Row Minimum: -1, -2 Maximin: -1 Column Maximum: 3 -1 Minimax: -1. 12/16/2021 Strictly Determined: because both players have selected their choice regardless of what the opponents does. In this case Tails for Tina and Heads for Sol, or the -1 element. The saddle amount is therefore -1. 10

Tina & Sol’s Coin Match Game: Strategy 12/16/2021 Since the payoffs are the same. The game is strictly determined. The best strategy for Tina is Tails and for Sol is Heads. The saddle point is -1: Sol can expect to loss 1 coin per match. Tina expects to win at least 1 coin per match. 11

Dominance When players have more than 2 strategies it becomes more difficult to analyze. We use dominance to help us. 12/16/2021 12

Dominance Suppose we examine 1 st 2 rows carefully, 2 always dominates 1 as 2 always dominates 3, both can be eliminated 12/16/2021 13

Dominance When looking at the columns, we think opposite. Here we will eliminate columns 1 and 3 12/16/2021 14

Dominance Once these strategies have been eliminated, we will use minimax and maximin Maximin- 100, minimax – 100 12/16/2021 15

Dominance Game is strictly determined with saddle point of 100. Dino’s best strategy is to offer free min, Sal’s is the same Dino’s will gain $100 over Sal’s using this strategy. 12/16/2021 16

Sol’s Revenge Sol’s new rules: Sol wins a coin if both are tails Sol losses 2 coins if they don’t match. 12/16/2021 17

Sol’s New Rules The game is no longer strictly determined. This means we need a mixture of heads and tails! 12/16/2021 18

Probability Distribution Outcome HH TT HT TH Amount 3 1 -2 -2 Probability . 25 Sol’s expectation: Tina’s expectation: 12/16/2021 19

Sol’s half and half 12/16/2021 20

Sol’s 60% 12/16/2021 21

Generalize 12/16/2021 22

Equalize 12/16/2021 23

Tina’s 12/16/2021 24

Sol’s Expectation 12/16/2021 25

Tina’s Expectation 12/16/2021 26