# 1 Game Theory. 2 Definitions Game theory -- formal way to analyze interactions among a group of rational agents behaving strategically Agents – players.

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1 Game Theory

2 Definitions Game theory -- formal way to analyze interactions among a group of rational agents behaving strategically Agents – players in the game Strategic behavior: account for interdependence in behavior (I know that you know that I know…..)

3 Rational – When accounting for interdependence, each player chooses their best action

4 Some simple examples Go into restaurant with a group of people. –Situation 1: everyone pays for their own food –Situation 2: divide the bill evenly Ultimatum game –\$100 is on the table, Player 1 splits the pot between players 1 and 2, player 2 accepts or rejects the offer You can bid on a jar of silver dollars. Unknown amount of dollars in the jar, highest bid wins the jar

5 Strategy Complete set of contingent plans for playing the game Detailed book on chess rather than just one move In simple games we consider, strategy is one move – so one decision or action is the strategy

6 Games can vary By move –Simultaneous or sequential Payoffs –Zero sum or non zero sum Players –2 or many

7 Simple game 2 players 2 potential moves 4 potential outcomes Great for pedagogical reasons, but, also many real world examples Can easily outline the players, strategies and payoffs in a simple 2 x 2 matrix

8 Firm B Option COption D Firm A Option Cπ ac,π bc π ac,π bd Option Dπ ad,π bc π ad,π bd Player on horizontal is 1 st Payoff in the pair

9 New toy Cost to firm is \$5 Two firms will sell Can sell the toy at a high (\$20/toy) or low (\$10/toy) price If prices are the same, firms split the market. If you are the low price firm, you get the lions share of the market and hence profits

10 P1P1 \$10\$20\$1020 Q1Q1 3015600 Profits 1 \$150\$225\$300\$0 P2P2 \$10\$2020\$20 Q2Q2 3025060 Profits 2 \$150\$225\$0\$300

11 Firm Pricing Payoff Matrix Toys R Us Price HighPrice Low Wal-Mart Price High225,2250,300 Price Low300,0150,150

12 Zero Sum Game Game of conflict – what you win, I lose Payoffs in any situation sum to zero Rock/paper/scissors You win, you get a \$1, I win, I get a \$1, tie no money is exchanges

13 Player B RockPaperScissor Rock0,0-1, 1 1, -1 Player APaper1, -10,0-1, 1 Scissor-1, 11, -10,0

14 Dominance Dominant strategy – a strategy that work at least as well as any other one, no matter what the other player does –If one exists, will play the dominant strategy… –But, none may exist To solve the game, eliminate dominated strategies Key assumption – player is acting in own self interest – doing the same thing you are doing

15 Toys R Us Price HighPrice Low Wal-Mart Price High225,2250,300 Price Low300,0150,150 Does Wal-Mart have a dominant strategy? Hold TRU action at “Price low”, what is best decision? Hold TRU decision at Price High, what is best you can do?

16 Dominant strategy equilibrium – where each player follows a dominant strategy

17 Player B RockPaperScissor Rock0,0-1, 1 1, -1 Player APaper1, -10,0-1, 1 Scissor-1, 11, -10,0 Does A have a dominant Strategy?

18 Player 2 LeftMiddleRight Player 1 Up2,21,14,0 Down1,24,13,5

19 Player 2 LeftMiddleRight Player 1 Up2,21,14,0 Down1,24,13,5

20 Look at player 2. In this case, M is a dominated strategy. If 1plays Up, Left is optimal. If 1 plays down, Right is optimal Now consider player 1. He knows player 2 will never take Middle so this is irrelevant For player 1, up is a dominant strategy Player 2 knows 1 will play up. Therefore 2’s only option is 2

21 Prisoner’s dilemma 2 suspects thought to be involved in a crime are arrested Have enough evidence for a minor conviction, but police know they were involved in more major crimes Put in separate rooms Each suspect is offered a deal. –If they turn states evidence, get reduced sentence –If the other person turns states evidence, you get hung out to dry

22 Prisoner’s Dilemma Prisoner B QuietBetrays Prisoner A Quiet-3,-3-10,-1 Betrays-1,-10-5,-5

23 Is there a Dominant Strategy? Prisoner B QuietBetrays Prisoner A Quiet-3,-3-10,-1 Betrays-1,-10-5,-5

24 Notice 2 things Although to betray is a dominant strategy, both parties could do better by keeping quiet Therefore, the dominant strategy does not necessarily generate the optimal outcome

25 Some games do not have dominant strategies You will need a different solution concept

26 Game of Chicken Buzz Gunderson SwerveDon’t Jim Stark Swerve0,0-5, 5 Don’t5, -5-10,-10

27 Game of Chicken Buzz Gunderson SwerveDon’t Jim Stark Swerve0,0-5, 5 Don’t5, -5-10,-10

28 Nash Equilibrium No player has an incentive to deviate from equilibrium play because their strategy is best response to her belief about the other player’s strategy Way to evaluate equilibrium or outcome of game All dominant strategies are Nash Equilibriums but not vice versa

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30 Player 2 LeftMiddleRight Player 1 Up2,21,14,0 Down1,24,13,5

31 Player 2 LeftMiddleRight Player 1 Up2,21,14,0 Down1,24,13,5

32 Prisoner’s Dilemma Prisoner B QuietBetrays Prisoner A Quiet-3,-3-10,-1 Betrays-1,-10-5,-5

34 Game of Chicken Buzz Gunderson SwerveDon’t James Dean Swerve0,0-5, 5 Don’t5, -5-10,-10

35 Battle of the Sexes Male/female want to go to the movies –He prefers Die Hard IV –She prefers Becoming Jane –They prefer to go together How should they decide what move to go to?

36 Battle of the Sexes Boyfriend Die Hard IV Becoming Jane Girlfriend Die Hard IV 5,100,0 Becoming Jane 2,210,5

37 Battle of the Sexes Boyfriend Die Hard IV Becoming Jane Girlfriend Die Hard IV 5,100,0 Becoming Jane 2,210,5

38 Nuclear Strategy Two super powers with incredible nuclear arsenal At any point in time, you can strike first or not Suppose that in a first strike, the nuclear arsenal of the attacked will be destroyed

39 No second strike capability Russia First StrikeNot US First Strike -10,-100,-8 Not-8,00,0

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41 Nuclear Deterrence Russia First StrikeNot US First Strike -10,-10-9,-8 Not-8,-90,0

42 Cartel Pricing Two oil produces Each can produce either 2 or 4 million barrels per day Iran can produce output at \$2/barrel Iraq can produce at \$4/barrell World output will be 6, 6 or 8 million barrels/day Price/barrel in these three scenarios would be \$25, \$15, and \$10

43 Iran’s profits with low production π =revenues – costs = \$25(2) – 2(2) = \$46 million Iraq’s profits w/ low production π = \$25(2) – 4(2) = \$42 million

44 Cartel Pricing Iraq Produce Low Produce High Iran Produce Low 46, 4226, 44 Produce High 52,2232, 24

45 Cartel Pricing Iraq Produce Low Produce High Iran Produce Low 46, 4226, 44 Produce High 52,2232, 24

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