So... What’s Game Theory?

1. 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.

2. 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)

3. 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)

4. 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.

5. 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

6. 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

7. 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?

8. 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

9. 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?

10. 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

11. Pirate Game: A Sample Complex Analysis • Work backwards • Pirate 4 – Needs 0 extra votes • Distribution is {0,0,0,100,0}

12. 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}

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}

14. 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}

15. 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}

16. 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!

17. Stag Hunt: The Payoff • 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.