Game theory Logic and Decision Making Unit 2
Introduction Game theory studies decisions in which the outcome depends (at least partly) on what other people do All decision makers are assumed to possess a common knowledge of rationality (or CKR), i.e. they do what is their best and know all the other players do their best
The Prisoner s Dilemma It is a class of games of the same type Trespassing? Robbery?
The potential deal You ll get 2 years for trespassing. But if you confess the attempted robbery you ll get 1 year. You can deny it but if your mate confesses you ll get 20 years! If you both confess, you ll get 10 years. What is your decision? How can I minimize the penalty??
You ll get 2 years for trespassing. But if you confess the attempted robbery you ll get 1 year. You can deny it but if your mate confesses you ll get 20 years! If you both deny, you ll get 10 years. The potential deal
PD s payoff matrix AL as a group BOB Strict Dominance Principle: one strategy strictly dominates another strategy when, regardless of what the other player does, the Tirst strategy always gives a better payoff Al s point of view Bob s point of view Group s point of view confess confess (deny, deny) individually
Can confessing be avoided? Maybe the other prisoner is not fully rational It does not change the situation: individually, it is better to confess Are the prisoners able to communicate each other so that they can coordinate their strategies? After they communicate, both would say: I m rational, not moral; confessing is better for myself. Then, it is still better to confess Under the assumption of common rationality, also communication is supertluous
PD in business: example 1 PD is much more than an artiticial example Example 1: two companies have to decide to set prices high or low. Each company s protits depends on what the other company decides. What is it rational to do? Whole market: 2000; high price: $100; low price: $60 as a group individually
Example 2: Advertise or not? Two companies has to decide whether to advertise their products. Advertising make them conquer a +40% market share, but if they both advertise they remain with the same share. What is it rational to do? Suppose whole market: 200; advertising cost: 20; market share through advertising: +40 as a group individually
Example 3: Pollute or not? Some chemical industries located near a river has to decide whether to limit their pollution into the river, thus making a lower protit, or ignore environmental concerns and maximise their protit. Each Tirm knows it is virtually risk- free to pollute; the river will get dirty if and only if the large majority of Tirms decide to pollute Any Tirm s payoff matrix Each =irm, individually As a group: do not pollute
Example 4: Public transport or car? Everyday the citizens of large cities have to decide whether to use public transportation or their cars. Car is more comfortable, but if anyone decides the same there is a huge traftic jam. What is it rational to do? As a group: less comfort and no traf=ic jam Any citizen s payoff matrix Each citizen, individually
Application of PD If companies act rationally, they end up choosing not the optimal outcome for the group This explains also why companies try to agree in secret to set prices high, although government authorities and consumer organisations do their best to prevent such deals (including legal rules preventing cartels)
PD s Generalized Payoff matrix Reward Sucker Temptation Punishment Symmetric, with T > R > P > S
Taxonomy of games PD is a non- cooperative, simultaneous- strategy, symmetric, nonzero- sum, two- person, Tinite game Zero- sum games: one player wins exactly as much as the other opponents lose (casino games, chess) Cooperative games: the players are able to form (or forced to keep) binding agreements Simultaneous- move games: the players decide their strategies without knowing what the other players will do (paper- stone- scissors). Otherwise, games are called sequential (and can be with perfect information or not) Symmetric games: all the players face the same strategies and outcomes N- person games: the number of players is N Non- iterated games: are played only once. Otherwise iterated
Dominance reasoning One strategy A dominates strictly B iff A s outcomes are always higher than B Some games can be solved through dominance reasoning (by iteratively deleting dominating strategies), given this assumptions Every player is rational Every player knows that every player is rational Every player knows that every player knows that every player is rational Every player knows that every player knows that every player knows ad intinitum Dominance principle is a valid rational principle
Examples with 2 players Among C i s none dominates R 2 R 1 ( delete R 1 ) C 3 C 2 C 1 ( delete C 1, C 2 ) R 2 R 3 ( delete R 3 ) Then, the pair (R 2, C 3 ) is the solution. Among R i s none dominates C 2 C 3 C 1 ( delete C 3, C 1 ) R 2 R 1 R 3 ( delete R 1, R 3 ) Then (R 2, C 2 ) is the solution.
Examples with 3 players 3rd player chooses X 2nd player 3rd player chooses Y 2nd player 1st player B A ( delete (A, *, *)) F G ( delete (*, G, *)) X Y ( delete (*, *, Y)) Then, (B, F, X) is the solution. Does it matter which strategy do you start deleting Tirst? NO
The centipede problem (1,0) Ann (0,2) Ben (3,1) Ann (2,4) Backward reasoning Ben Ann should stop with the pot (1,0) It is not a problem with dominance (same result reached with other methods), but with the rules (3,3)
Dominance reasoning 1) Dominance principle (strict dominance) 2) Common Knowledge of Rationality Among C i s none dominates R 2 R 1 ( delete R 1 ) C 3 C 2 C 1 ( delete C 1, C 2 ) R 2 R 3 ( delete R 3 ) Then, the pair (R 2, C 3 ) is the solution.
2- person zero- sum games The matrix suftices to contain the outcomes of just one player. Solutions can be obtained Sometimes by dominance Sometimes by searching for equilibrium strategies: a set of strategies, one for each player, such that no player has incentive to change his/her strategy given what the other players are doing Given (R3,C2) none could reach a better outcome by unilaterally switching to another strategy
Nash Equilibrium A Nash equilibrium is a stable strategy in which no participant can gain by changing the strategy as long as the other participants remain unchanged Thus, in a Nash equilibrium the strategy maximizes the payoffs of each player, given that the strategies of the others are held Tixed; this entails that no player has incentive to change his/her strategy given what the other players are doing The idea https://www.youtube.com/watch?v=5adpymgvzfo
Finding for equilibria Minimax condition (it is sufticient, not necessary): if the outcomes equals the minimal value of the row and the maximal value of the column, then that pair of strategies is in equilibrium
Two kind of strategies What should R and C choose? A pure strategy provides a complete detinition of how a player will play. It determines the move for any situation Heads and Tails are pure strategies for both players A mixed strategy is an assignment of a probability to each pure strategy. This allows to randomly select a pure strategy (½ Heads, ½ Tails) is a mixed strategy for both players (¼ Heads, ¾ Tails) is a mixed strategy for both players Matching pennies game
Nash Equilibria in Mixed strategies If a player plays more than one strategy with strictly positive probability, then he must be indifferent between the strategies he plays with strictly positive probability (½ Heads, ½ Tails) means the player is indifferent between playing Heads with probability ½ and playing Tails with prob. ½ A mixed strategy is a mixed strategy Nash equilibrium iff every player has no incentive to move to one of the pure strategies he is mixing given that the other players remain unchanged Put simply, the EU of the single choices combined in the mixed strategy should be equal. This allow to determine what probability should be assigned
Understanding mixed trategies Consider player 1 (Rows). If player 2 plays (¼ Heads, ¾ Tails), what should player 1 do? EU(H)=¼(1)+¾(- 1)=- 2/4 EU(T)=¼(- 1)+¾(1)=2/4 à play Tails 1/4 3/4 Is there a set of strategies that are in equilibrium? Are there values of probabilities so that player 1 is indifferent between the strategies which are mixed randomly?
Example Consider Tirst player 1. Suppose player 2 plays (p Heads, 1- p Tails) then, imposing the EUs to be equal: EU(H) = (1)p+(- 1)(1- p) = - 1+2p EU(T) = (- 1)p+(1)(1- p) = 1-2p EU(H) = EU(T) à - 1+2p = 1-2p à p=½ is the probability that, if adopted by player 2, makes player 1 s move indifferent between the Heads and Tails The same holds for player 2. Then ((½ Heads, ½ Tails), (½ Heads, ½ Tails)) is in equibrium p 1- p
The Minimax Theorem Minimax theorem. Every 2- person zero- sum game has a solution, i.e. a pair of strategies in equilibrium. If there is more than one pair, they all have the same expected utility. Thus, every 2- person zero- sum game has a solution. It can be given by pure or mixed strategies.
The Prisoner Dilemma This is a 2- player nonzero- sum game Player R: R2 dominates R1 Player C: C2 dominates C1 à (R2, C2) is a Nash equilibrium Are there mixed Nash equilibria? Consider Tirst player R. Suppose p(c1)=p, then EU(R1)=(- 2)p+(- 20)(1- p)=18p- 20, EU(R2)= =9p- 10 hence EU(R1)=EU(R2) à p=10/9 The same holds for player C. à No other Nash equilibrium
Lessons from the PD It illustrates the contlicts between individual and group rationality where cooperation and competition is involved (in business as in politics, or in other social settings) It shows that when each individual pursues his/her own self- interest, the outcome is worse than if they had both cooperated Connection to the Tragedy of the Commons (when a shared resource is overused by the people using it, even though they know that if they all overuse it then it will run out) and the problem of free riding
The Stag Hunt This is a 2- player nonzero- sum game Dominance principle does not help Using Minimax condition à (R1, C1) and (R2, C2) are Nash equilibria Are there mixed Nash equilibria? Consider Tirst for R. Suppose p(c1)=p, then EU(R1)=25p+0(1- p)=25p, EU(R2)=5p+5(1- p)=5 hence EU(R1)=EU(R2) à p=1/5 The same holds for C à ((1/5 R1, 4/5 R2), (1/5 C1, 4/5 C2)) is a Nash equilibrium
The Stag Hunt and then? We have 3 possible solutions. Thus some strategies have been eliminated But, between the three, what should we expect from a rational agent to do? It is reasonable to treat it as a decision under risk, with a symmetric reasoning (holds for both R and C) Taking for granted that we maximise EU, if R has a subjective probability p>1/5 that C will hunt for stag, then also R should hunt for stag If R is risk adverse, R should hunt for hare
Lessons from the Stag Hunt Avoid risk aversion: in order to reach an outcome that is the best for the group, players have to be prepared to take at least some moderated risks Trust and Cooperate: If people trust each other the group will have outcomes that are better for the group as a whole and for every individual the benetits of trust always run the risk of being destroyed by risk- averse DMs we should always preserve and increase the degree of trust Otherwise, if we are too risk- averse or do not trust our fellows will cooperate, we will not reach the best outcome
The Battle of sexes This is a 2- player nonzero- sum game Using Minimax condition à (R1, C1) and (R2, C2) are Nash equilibria Are there mixed Nash equilibria? Suppose p(c1)=p, then EU(R1)=EU(R2) à p=1/3 Suppose p(r1)=q, then EU(C1)=EU(C2) à q=2/3 à ((2/3 R1, 1/3 R2), (1/3 C1, 2/3 C2)) is a Nash equilibrium But what should they end up doing?
The Battle of sexes and then? We have 3 possible solutions. What should we do? Rationally R would prefer R1, C would prefer C2 à leaving both unsatistied If they both are not seltish, R would prefer R2, C would prefer C1 à leaving both unsatistied It seems there is no solution Again, if you are able to assign probabilities (subjectively) it can be treated as a decision under risk Otherwise, if you can t (as on the Tirst date) there isn t agreement in the literature
The chicken game https://www.youtube.com/watch?v=u7hz9jkrwvo Rebel without a cause This is a 2- player nonzero- sum game Using Minimax condition à (R1, C2) and (R2, C1) are Nash equilibria Suppose p(c1)=p, then EU(R1)=(- 100)p+1(1- p)=eu(r2)=(- 1)p+0(1- p) à p=1/100 Suppose p(r1)=q, then EU(C1)=(- 100)q+1(1- q)=eu(c2)=(- 1)q+0(1- q) à q=1/100 à ((1/100 R1, 99/100 R2), (1/100 C1, 99/100 C2)) is a Nash equilibrium But what should they end up doing?
The bargaining problem R and C can split $100. They have to write an amount: if the amount sum to more than $100, they will get nothing; otherwise, each player gets the amount written. They can communicate and form any binding agreement. This is a 2- player cooperative game that models bargaining situations; in essence it is an equilibrium selection problem Every possible split of $100 is a Nash equilibrium
The Bargaining problem (cont d) Nash proposal: rational players facing the bargaining problem would agree on 4 axioms Nash equilibrium: whatever rational players agree on, the chosen strategy will be a Nash equilibrium Irrelevance to utility representation: the chosen strategies should be invariant to the choice of vnm utility function representing players preferences Independence of irrelevant alternative: the solution does not depend on whether irrelevant alternative are added or removed to the set of alternatives Symmetry: The strategies of the players will be identical iff their utility functions are identical
The Bargaining problem (cont d) By assuming the 4 axioms, Nash s solution is this: if the utility functions of R and C are u(x) and v(x), then rational player would always agree to divide the money so that u(x) v(100- x) is maximised this entails the bargaining problem has just 1 solution, despite there are intinitely many Nash equilibria If each player s utility is directly proportional to the money they get (so that u(x)=x and v(x)=x), then x(100- x) is maximised for x=50
Application to ethics Ethical principle: any agreement made between rational individuals is morally acceptable If we agree on this principle (which is a bridge- premise between morality and rationality), any ethical problem can be reduced to a problem of rationality Using this principle, all problems of distributive ethics can be solved assuming Nash s axioms Problems: what agreed from rational individuals may have negative consequences for others According to the ethicist perspective, behaviors morally good are not concerned with maximising consequences
Iterated games Two kinds: =initely iterated games, where the game is played a Tinite number of times; players know when they are playing the last iteration in=initely iterated games, where the game is played an intinite number of times, or also a Tinite number but players do not know when they are playing the last iteration In iterated games it is important to consider how players may react to a move of the opponent in the next round
Finitely iterated PD In Tinitely iterated prisoner s dilemma rational players should behave exactly as in the one- shot version (confess). Why? Backward induction argument Last round: both players cannot get rewards from cooperation in the next round, so they confess Penultimate round: players cannot get rewards as in the Tinal round they know each player is better confessing. Thus they confess Players end up with confessing all the rounds
InTinitely iterated PD In intinitely iterated PD the backward induction argument fails, because there is not the last round to start the argument. Then if a player confesses, he must take into account how the opponent will punish him in the next round. So it would be better to cooperate but if the opponent does not cooperate, then it would be better to confess (so the players must observe and learn each other s behavior) TIT for TAT strategy: cooperate in the Tirst round, then adjust the behavior to whatever strategy the opponent did in the previous round. But it is not the only strategy If both play TfT then they reach a NE But also if they always confess they reach a NE
The Student Game Be careful: we are now really going to play this game. You have the opportunity to earn some extra credit (this is a real game) on your Tinal paper grade. Select whether you want 1 point or 3 points added onto your Tinal 2 nd homework grade. But there s a small catch: if more of 10% of the class select 3 points, then no one gets any points. Your responses will be anonymous to the rest of the class.