Games What is ‘Game Theory’? There are several tools and techniques used by applied modelers to generate testable hypotheses Modeling techniques widely used in applied economic work: 1) constrained optimization 2) comparative statics 3) non-cooperative game theory What is the added value of a model? The value added is to formulate hypotheses, make assumption clear, to be precise, to discipline the analysis General Games A game is a formal representation of a situation in which a number of individuals interact in a setting of strategic interdependence

A game has the following features 1)A set of players - description of who is involved 2)Rules of the game - an order of play, who moves when - description of the information available to each player at every point in the game, what they know when they move - a set of actions for each player, what they can do 3) Outcomes - a set of outcomes that depend on the sequence of actions taken by each player - for each possible set of actions by the players what is the outcome of the game 4) Payoffs - a description of the players’ preferences over possible outcomes

Strategy Defn: A strategy is a complete contingent plan, or decision rule, that specifies how the player will act in every possible distinguishable circumstance in which he might be called upon to move. A strategy is distinct from an action; only in the simplest games are they the same The set of circumstances is represented by her collection of information sets, with each info set representing a different distinguishable circumstance in which she may need to move A player’s strategy specifies how she plans to move at each one of her information sets, should it be reached during play of the game

Non-cooperative game theory : Equilibrium is defined in terms of strategies; it extends single agent optimization to multiple agents that interact, so that many players are optimizing or attempting to optimize simultaneously. We can make a distinction between games of: Complete information – all players know the payoffs of all others players (they know each others utility functions, or preferences) Incomplete information – some of the players do not know some of the payoffs of the other players A finite # of strategies -- limited number of choices An infinite # of strategies – continuous choice variable such as price or quantity; we can use calculus

Simultaneous Move Games Static games in which all players move once and they all move at the same time. We assume that players are - rational (they optimize) - fully knowledgeable about the structure of the game (including each other’s strategy sets) - each other’s rationality (other players optimize by maximizing utility).

Simultaneous Move Games

Three components of the Normal Form 1) Set of players – who are they? 2) Strategy sets – what can they do? 3) Utility functions defined on strategies – what are their preferences?

Critical Concept Best response functions are an essential concept in order to understand and solve non-cooperative games.

Dominance

Payoff Matrix These can be used to represent player’s payoffs given different outcomes Suppose 2 major TV networks each have a great show; each needs to pick the time to run the show, and the show runs for 2 hours To keep it simple, suppose the choice is between 8 P.M. And 9 P.M. Each network is trying to maximize its number of viewers, where the number of viewers is measured in millions Network 2 Network 1 8 P.M.9 P.M. 8 P.M.36, 3339, 28 9 P.M.30, 3632, 30

Dominant strategies If both run their show at 8 P.M. then network (1) gets 36 million viewers network (2) gets 33 million viewers, etc… Now to solve the game note that no matter what (2) does (1) maximizes the # of viewers by running the show at 8 P.M. (1) has a dominant strategy – a best response to any strategy that the other player might pick (2) also has a dominant strategy – run the show at 8 P.M. So the unique outcome, solution, prediction is that both run their shows at 8 P.M. The TV example is easy to solve because both players have dominant strategies; Each can essentially ignore what the other is doing

Clothing store Sometimes you might not have a dominant strategy but if you know your opponent’s payoffs then you can see how you should play Ex: suppose you run a clothing store, store (1); you have a competitor, store (2) - you know your competitor can either promote girl’s or women’s clothes - you can either promote girl’s clothes or children’s clothes - you come up with the following payoff table Store 2 Store 1 Girl’s clothesWomen’s clothes Girl’s clothes0, 04, 2 Women’s clothes2, 22, 4

Clothing store Now store (1) does not have a dominant strategy - if (2) promotes girl’s clothes (1) should promote children’s - if (2) promotes women’s (1) should promote girl’s How can (1) figure out what to do? Recognize that (2) has a dominant strategy – promote women’s clothes - given this, (1) should promote girl’s clothes Store 2 Store 1 Girl’s clothesWomen’s clothes Girl’s clothes0, 04, 2 Women’s clothes2, 22, 4

Prisoner’s dilemma Both players play their dominant strategy, and as a result are worse off then they would be if they could avoid playing the dominant strategy. 2 prisoners; separate rooms; if both refuse to talk then they both get light sentences if one refuses to talk and the other talks, the one who talks goes free and the other gets a heavy sentence if both talk they both get medium sentences 2 1 CooperateDefect Cooperate10, 105, 12 Defect12, 57, 7 Both would like to cooperate but both defect and are worse off

Iterated Deletion of Dominated Strategies Sometimes eliminating dominated strategies allows you to solve the game Sumo wrestling: Does sumo wrestling have points? These expected payoffs may take into account ego points for winning in a particular way or avoiding injury 2 1 run straightfake leftfake right run straight5, 52, 24, 1 fake left7, 32, 40, 3 fake right6, 23, 30, 0 For (2), the strategy fake right is dominated by fake left Given that (2) never fakes right, running straight is dominated for (1) Given that (1) never runs straight, (2) never runs straight Given that (2) always fakes left, (1) always fakes right