Slide 1 of 13 So... What’s Game Theory? Game theory refers to a branch of applied math that deals with the strategic interactions between various ‘agents’, or players. Agents choose strategies based on information about the strategies of other players in an attempt to maximize the return of their actions. Provides a formal model for economic social situations that involve the interactions of two or more parties in negotiations.

Slide 2 of 13 Let’s Lay Down Some Framework… Simultaneous vs. Sequential: Simultaneous games are those in which players act without knowledge of previous actions or outcomes. Sequential games refer to those in which players have some knowledge of previous actions Perfect Information: all players know the moves previously made by all other players (examples include chess, go, and mancala)

Slide 3 of 13 Games Can Get Funky Symmetric Games: Payoffs depend only on the strategies employed, not on the players employing them (examples: prisoner’s dilema, chicken) Asymmetric: non-identical strategy sets for each player (example: dictator game) Zero Sum Games: applicable to real-life situations; players can neither increase nor decrease available resources, but rather benefit at the expense of other players (examples: poker, chess) Non-zero games: combinations of strategies have a net sum that doesn’t = zero; one player’s gain does not necessarily correspond with another’s loss (example: prisoner’s dilemma)

Slide 4 of 13 Nash Equilibrium Solution concept for a game involving two or more players No player benefits from changing his or her strategy while other strategies remain unchanges Player A is making the best decision A can, taking into account Player B's decision, and B is making the best decision B can, taking into account A's decision.

Slide 5 of 13 How are Games Represented? Normal Form (Matrix) Used for games with only one decision by each player N player game = N-Dimensional matrix, with the payoffs for each player in the cell To the right, general matrix for a 2- player, 2-strategy game (Player 1) Strategy A (Player 1) Strategy B (Player 2) Strategy AP(A,A)P(B,A) (Player 2) Strategy BP(A,B)P(B,B)

Slide 6 of 13 How are Games Represented? Extensive Form (Tree) Used when order of decision is important or when each player has more than one move Used for many commonly known games  Chess  Checkers  Go  Tic-Tac-Toe Example 2-player, 2-branching tree on right

Slide 7 of 13 Prisoner’s Dilemma - Intro Two criminals can turn evidence on the other or remain silent If they both cooperate (remain silent), they each only get 1 year If one cooperates and one defects (turns evidence), the one that defected gets 0 years and the one that cooperated gets 5 years If they both defect, they each get 3 years What is the best strategy? (Player 1) Cooperate (Player 1) Defect (Player 2) Cooperate1,10,5 (Player 2) Defect0,53,3

Slide 8 of 13 Prisoner’s Dilemma - Analysis If other player cooperates, defecting gives less time in jail (0 vs. 1) If other player defects, defecting gives less time in jail (3 vs. 5) Defecting is a dominant strategy (best no matter what) What does this mean? For any single game, defection is the best option If the games are iterative, but independent (i.e. it is impossible for your strategy in one game affect your opponent’s strategy in another game), defection is the best option If games are iterative, but not independent?  Becomes a tree-based, multi-move game to which this analysis no longer applies (Player 1) Cooperate (Player 1) Defect (Player 2) Cooperate1,10,5 (Player 2) Defect0,53,3

Slide 9 of 13 Pirate Game: A Sample Complex Analysis A group of pirates splits up loot in the following manner Pirates are ranked #1 through #N by seniority Pirate 1 makes a proposal for splitting gold, which is voted on (“Yes” or “No”) by all pirates If at least half vote yes, proposal passes Otherwise, Pirate 1 is thrown overboard, everyone moves up a rank, and the process repeats Assuming pirates are rational and wish to maximize, in order of importance, survival, wealth, and rank, how would 5 pirates split up 100 gold pieces?

Slide 10 of 13 Pirate Game: A Sample Complex Analysis Of the proposals by pirate 1 which pass, which maximizes his utility? For at least 2 other pirates, utility from “Yes” must be more than utility from “No” Utility from “Yes” can be set by pirate 1 What is utility for pirates 2-5 for saying “No”?  Utility for those pirates from Pirate 2’s proposal

Slide 11 of 13 Pirate Game: A Sample Complex Analysis Work backwards Pirate 4 – Needs 0 extra votes  Distribution is {0,0,0,100,0}

Slide 12 of 13 Pirate Game: A Sample Complex Analysis Work backwards Pirate 4 – Needs 0 extra votes  Distribution is {0,0,0,100,0} Pirate 3 – Needs 1 extra vote  Can buy vote from 5 for 1 gold  Distribution is {0,0,99,0,1}

Slide 13 of 13 Pirate Game: A Sample Complex Analysis Work backwards Pirate 4 – Needs 0 extra votes  Distribution is {0,0,0,100,0} Pirate 3 – Needs 1 extra vote  Can buy vote from 5 for 1 gold  Distribution is {0,0,99,0,1} Pirate 2 – Needs 1 extra vote  Can buy vote from 4 for 1 gold  Distribution is {0,99,0,1,0}

Slide 14 of 13 Pirate Game: A Sample Complex Analysis Work backwards Pirate 4 – Needs 0 extra votes  Distribution is {0,0,0,100,0} Pirate 3 – Needs 1 extra vote  Can buy vote from 5 for 1 gold  Distribution is {0,0,99,0,1} Pirate 2 – Needs 1 extra vote  Can buy vote from 4 for 1 gold  Distribution is {0,99,0,1,0} Pirate 1 – Needs 2 extra votes  Can buy vote from 3 and 5 for 1 each  Distribution is {98,0,1,0,1}

Slide 15 of 13 Pirate Game: A Sample Complex Analysis Work backwards Pirate 4 – Needs 0 extra votes  Distribution is {0,0,0,100,0} Pirate 3 – Needs 1 extra vote  Can buy vote from 5 for 1 gold  Distribution is {0,0,99,0,1} Pirate 2 – Needs 1 extra vote  Can buy vote from 4 for 1 gold  Distribution is {0,99,0,1,0} Pirate 1 – Needs 2 extra votes  Can buy vote from 3 and 5 for 1 each  Distribution is {98,0,1,0,1} Final answer: {98,0,1,0,1}

Slide 16 of 13 The Stag Hunt Conflict modeling benefits of social cooperation: suggested by Rousseau as a model for the social contract Each player can choose to hunt a hare or a stag. If he hunts a stag, he must have the cooperation of his partner in order to succeed. An individual can get a hare by himself, but a hare is worth less than a stag Let’s examine a payoff matrix!

Slide 17 of 13 Stag Hunt: The Payoff StagHare Stag4,41,3 Hare3,13,3 Let a>b>or=d>c While each player choosing a hare would represent a Nash Equilibrium, it would not represent the maximum payoff Interestingly, the greatest payoff results from mutual cooperation between the players. This requires, however, an element of trust between members of society. StagHare StagA, aC, b HareB, cD, d