Introduction to Game Theory Part 1. Static games of complete information Chapter 1. Normal form games and Nash equilibrium Ciclo Profissional 2 o Semestre / 2011 Graduação em Ciências Econômicas V. Filipe Martins-da-Rocha (FGV) Introduction to Game Theory August, 2011 1 / 31
Topics covered 1 Normal-form representation of games 2 Iterated elimination of strictly dominated strategies 3 Motivation and definition of Nash equilibrium V. Filipe Martins-da-Rocha (FGV) Introduction to Game Theory August, 2011 2 / 31
What is a game? Definition A game is a formal representation of a situation in which a number of individuals interact in a setting of strategic interdependence Each individual s welfare depends not only on his own actions but also on the actions of the other individuals The actions that are best for an individual to take may depend on what he expects the other players to do V. Filipe Martins-da-Rocha (FGV) Introduction to Game Theory August, 2011 3 / 31
Normal-form representation of games In the normal-form representation of the game each player simultaneously chooses a strategy the combination of strategies chosen by players determines a payoff for each player Example The prisoners dilemma V. Filipe Martins-da-Rocha (FGV) Introduction to Game Theory August, 2011 4 / 31
The prisoners dilema: the environment Two suspects are arrested and charged with a crime The police lack sufficient evidence to convict the suspects, unless at least one confesses The suspects are in separate cells The police explain the consequences that will follow from the actions they could take V. Filipe Martins-da-Rocha (FGV) Introduction to Game Theory August, 2011 5 / 31
The prisoners dilema: actions and payoffs If neither confesses then will be convicted of a minor offense and sentenced to one month in jail If both confess then both will be sentenced to jail for six months If one confesses but the other does not, then the confessor will be released immediately but the other will be sentenced to nine months in jail Six for crime Three for obstructing justice V. Filipe Martins-da-Rocha (FGV) Introduction to Game Theory August, 2011 6 / 31
The prisoners dilema: matrix representation Each player has two strategies: Confess or Not confess We implicitly assume that each player does not like to stay in jail Prisoner i 1 Prisoner i 2 Not confess Confess Not confess 1, 1 9, 0 Confess 0, 9 6, 6 Prisoners dilemma V. Filipe Martins-da-Rocha (FGV) Introduction to Game Theory August, 2011 7 / 31
The normal form representation: Players A finite set I of players We write player i where i is the name of the player and I is the collection of names We denote by n the number of players, i.e., n = #I The set I may denoted by I = {1, 2,..., n} We prefer the notation I = {i 1, i 2,..., i n } V. Filipe Martins-da-Rocha (FGV) Introduction to Game Theory August, 2011 8 / 31
The normal form representation: Strategies The set of strategies available to player i is denoted by S i An element s i in S i is called a strategy (or play or action) The set S i is called strategy space and may have any structure: finite, countable, metric space, vector space The collection (s i ) i I = (s i1,..., s in ) is called a strategy profile and denoted by s or s Given an agent j and a profile s, we denote by (s j, s j) the new profile σ = (σ i ) i I defined by σ i = If j = i k for some 1 < k < n then { s j if i = j s i if i j (s j, s j ) = (s i1,..., s ik 1, s i k, s ik+1,..., s in ) V. Filipe Martins-da-Rocha (FGV) Introduction to Game Theory August, 2011 9 / 31
The normal form representation: Payoffs The payoff of player i is a function u i : j I S j [, + ] s u i (s) where u i (s) is the payoff of player i when he plays strategy s i and any other player j plays strategy s j We use alternatively the following notation u i (s) = u i ((s j ) j I ) = u i (s i, s i ) = u i (s i1, s i2,..., s in ) Sometimes, abusing notations we write u i (s 1, s 2,..., s n ) V. Filipe Martins-da-Rocha (FGV) Introduction to Game Theory August, 2011 10 / 31
The normal form representation Definition A game in normal form is a family G = (S i, u i ) i I where for each i I S i is a set u i is a function from S = k I S k to [, ] Question? We should know describe how to solve a game-theoretic problem Can we anticipate how a game will be played? What should we expect to observe in a game played by rational players who are fully knowledgeable about the structure of the game and each others rationality? V. Filipe Martins-da-Rocha (FGV) Introduction to Game Theory August, 2011 11 / 31
Simultaneous moves In a normal form game the players choose their strategies simultaneously This does not imply that they act simultaneously It suffices that each choose his or her action without knowledge of the others choices For the prisoners dilema, the prisoners may reach decisions at arbitrary times but it must be in separate cells Bidders in an sealed-bid auction V. Filipe Martins-da-Rocha (FGV) Introduction to Game Theory August, 2011 12 / 31
Strictly dominated strategies Definition Consider a normal form game (S i, u i ) i I Let s i and s i be two strategies in S i Strategy s i is strictly dominated by strategy s i if for each possible combination of the other players strategies, the player i s payoff from playing s i is strictly less than the payoff playing s i Formally, s i k i S k, u i (s i, s i ) < u i (s i, s i ) Rationality Rational players do not play strictly dominated strategies V. Filipe Martins-da-Rocha (FGV) Introduction to Game Theory August, 2011 13 / 31
Strictly dominated strategies: The prisoners dilemma For a prisoner, playing Not confess is strictly dominated by playing Confess Assume we are player i 1 If player i 2 chooses Confess We prefer to play Confess and stay 6 months in jail Then playing Not confess and stay 9 months in jail If player i 2 chooses Not confess We prefer to play Confess and be free Then playing Not confess and stay 1 month in jail Prisoner i 1 Prisoner i 2 Not confess Confess Not confess 1, 1 9, 0 Confess 0, 9 6, 6 Prisoners dilemma V. Filipe Martins-da-Rocha (FGV) Introduction to Game Theory August, 2011 14 / 31
Strictly dominated strategies: The prisoners dilemma A rational player will choose to play Confess The outcome reached by the two prisoners is (Confess,Confess) This results in a worse payoff for both players than would (Not confess,not confess) This inefficiency is a consequence of the lack of co-ordination This happens in many other situations the arms race the free-rider problem in the provision of public goods V. Filipe Martins-da-Rocha (FGV) Introduction to Game Theory August, 2011 15 / 31
Strictly dominated strategies: Iterated elimination Can we use the idea that rational players do not play strictly dominated strategies to find a solution to other games? Consider a game (in normal form) with two players Player i 1 has two available strategies S i1 = {Up, Down} Player i 2 has three available strategies S i2 = {Left, Middle, Right} The payoffs are given by the following matrix Player i 1 Player i 2 Left Middle Right Up 1, 0 1, 2 0, 1 Down 0, 3 0, 1 2, 0 V. Filipe Martins-da-Rocha (FGV) Introduction to Game Theory August, 2011 16 / 31
Strictly dominated strategies: Iterated elimination Player i 1 Player i 2 Left Middle Right Up 1, 0 1, 2 0, 1 Down 0, 3 0, 1 2, 0 For Player i 1 Up is not strictly dominated by Down Down is not strictly dominated by Up For Player i 2 the strategy Right is strictly dominated by Middle Player i 2 will never play Right If Player i 1 knows that Player i 2 is rational Then Player i 1 can eliminate Right from Player i 2 s strategy set V. Filipe Martins-da-Rocha (FGV) Introduction to Game Theory August, 2011 17 / 31
Strictly dominated strategies: Iterated elimination Both players can play the game as if it were the following game Player i 1 Player i 2 Left Middle Up 1, 0 1, 2 Down 0, 3 0, 1 For Player i 1 the strategy Down is strictly dominated by Up If Player i 2 knows that Player i 1 is rational And Player i 2 knows that Player i 1 knows that Player i 2 is rational Then Player i 2 can eliminate Down from Player i 1 s strategy space V. Filipe Martins-da-Rocha (FGV) Introduction to Game Theory August, 2011 18 / 31
Strictly dominated strategies: Iterated elimination Now the game is as follows Player i 1 Player i 2 Left Middle Up 1, 0 1, 2 For Player i 2 the strategy Left is strictly dominated by Middle By iterated elimination of strictly dominated strategies The outcome of the game is (Up,Middle) V. Filipe Martins-da-Rocha (FGV) Introduction to Game Theory August, 2011 19 / 31
Strictly dominated strategies: Iterated elimination Definition This process is called iterated elimination of strictly dominated strategies Proposition The set of strategy profiles that survive to iterated elimination of strictly dominated strategies is independent of the order of deletion V. Filipe Martins-da-Rocha (FGV) Introduction to Game Theory August, 2011 20 / 31
Strictly dominated strategies: Iterated elimination Drawbacks Each step requires a further assumption about what the players know about each other s rationality To apply the process for an arbitrary number of steps, we need to assume that it is common knowledge that players are rational All the players are rational All the players know that all the players are rational So on, ad infinitum This process often produces a very imprecise prediction about the play of the game V. Filipe Martins-da-Rocha (FGV) Introduction to Game Theory August, 2011 21 / 31
Strictly dominated strategies: Limitations Consider the following game L C R T 0, 4 4, 0 5, 3 M 4, 0 0, 4 5, 3 B 3, 5 3, 5 6, 6 There are no strictly dominated strategies to be eliminated The process produces no prediction whatsoever about the play of the game Question Is there a stronger solution concept than iterated elimination of strictly dominated strategies which produces much tighter predictions in a very broad class of games? V. Filipe Martins-da-Rocha (FGV) Introduction to Game Theory August, 2011 22 / 31
Nash equilibrium: Motivation Suppose that game theory makes a unique prediction about the strategy each player will choose In order for this prediction to be compatible with incentives (or correct) it is necessary that each player be willing to choose the strategy predicted by the theory Thus each player s predicted strategies must be that player s best response to the predicted strategies of other players Such a prediction could be called strategically stable or self-enforcing Because no single player wants to deviate from his or her predicted strategy A solution of the game satisfying the previous property is called a Nash equilibrium V. Filipe Martins-da-Rocha (FGV) Introduction to Game Theory August, 2011 23 / 31
Nash equilibrium: Definition Definition Consider a game G = (S i, u i ) i I A strategy profile s = (s ) i I is a Nash equilibrium of G if for each player i, the strategy s i is player i s best response to the strategies specified in s for the other players In other words, s = (s i ) i I is a Nash equilibrium if i I, s i argmax{u i (s i, s i) : s i S i } i.e., s i S i, u i (s i, s i) u i (s i, s i) Remark The set argmax{u i (s i, s i) : s i S i } may not be uniquely valued V. Filipe Martins-da-Rocha (FGV) Introduction to Game Theory August, 2011 24 / 31
Nash equilibrium: Interpretation If the theory offers the profile s = (s i ) i I that is not a Nash equilibrium then there exists at least one player that will have an incentive to deviate from the theory s prediction If a convention is to develop about how to play a given game then the strategies prescribed by the convention must be a Nash equilibrium, else at least one player will not abide the convention V. Filipe Martins-da-Rocha (FGV) Introduction to Game Theory August, 2011 25 / 31
Nash equilibrium: Examples In a two-player game we can compute the set of Nash equilibria as follows: For each player For each strategy for this player Determine the other player s best response to that strategy Underline the corresponding payoff on the matrix A pair of strategies (profile) is Nash equilibrium if both corresponding payoffs are underlined in the matrix L C R T 0, 4 4, 0 5, 3 M 4, 0 0, 4 5, 3 B 3, 5 3, 5 6, 6 V. Filipe Martins-da-Rocha (FGV) Introduction to Game Theory August, 2011 26 / 31
Nash equilibrium: Examples Player i 1 Player i 2 Left Middle Right Up 1, 0 1, 2 0, 1 Down 0, 3 0, 1 2, 0 Prisoner i 1 Prisoner i 2 Not confess Confess Not confess 1, 1 9, 0 Confess 0, 9 6, 6 Prisoners dilemma V. Filipe Martins-da-Rocha (FGV) Introduction to Game Theory August, 2011 27 / 31
Nash equilibrium: a stronger solution Consider a game G = (S i, u i ) i I Proposition If iterated elimination of strictly dominated strategies eliminates all but the strategy profile s = (s i ) i I then s is the unique Nash equilibrium of the game Theorem If the strategy profile s is a Nash equilibrium then s survives iterated elimination of strictly dominated strategies V. Filipe Martins-da-Rocha (FGV) Introduction to Game Theory August, 2011 28 / 31
Nash equilibrium: a stronger solution Remark Nash equilibrium is a stronger solution concept than iterated elimination of strictly dominated strategies Is it too strong? Can we be sure that a Nash equilibrium exists? There can be strategy profiles that survive iterated elimination of strictly dominated strategies but which are not Nash equilibria V. Filipe Martins-da-Rocha (FGV) Introduction to Game Theory August, 2011 29 / 31
A classic example: The battle of sexes A man (Pat) and a woman (Chris) are trying to decide on an evening s entertainment While at workplaces, Pat and Chris must choose to attend either the opera or a rock concert Both players would rather spend the evening together than apart Chris Pat Opera Rock Opera 2, 1 0, 0 Rock 0, 0 1, 2 V. Filipe Martins-da-Rocha (FGV) Introduction to Game Theory August, 2011 30 / 31
A classic example: The battle of sexes There are two Nash equilibria: (Opera,Opera) and (Rock,Rock) We will see that in some games with multiple Nash equilibria one equilibrium stands out as the compelling solution: in particular a convention can be developed In the example above, the Nash equilibrium concept loses much of its appeal as a prediction of play since both equilibria seem equally compelling: none can be developed as a convention V. Filipe Martins-da-Rocha (FGV) Introduction to Game Theory August, 2011 31 / 31