Game Theory

1. Game Theory M i c r o e c o n o m y João Castro Miguel Faria Sofia Taborda Cristina Carias Master in Engineering Policy and Management of Technology 24th February

2. Estratégia de Soares é minimizar Alegre “Mário Soares pretende ignorar tanto quanto possível a candidatura de Manuel Alegre...” “...garantiu que "não muda nada" na sua estratégia por causa de Alegre e repetiu que o seu adversário é "o candidato da direita, que não sei ainda se é, mas que espero que seja o Prof. Cavaco Silva". Diário de Noticias 26/09/2005 Sondagem inicial: Cavaco Silva 53,0% Mário Soares 16,9% Manuel Alegre 16, 2% Soares agita meios políticos E deixa Alegre fora da corrida a Belém. Sócrates afirmou preferir ex-Presidente da República. Cavaco não se deixa inibir. «PS ficou dividido», diz PSD. Alegre não comenta «reflexão» de Soares, que quer «escutar o sentimento» dos portugueses antes de avançar. Portugal Diário, 24/07/2005 Resultados: Cavaco Silva 50,6% Manuel Alegre 20,7% Mário Soares 14,3% José Sócrates proíbe represálias sobre Manuel Alegre “...o apoio a Alegre, se viesse a ocorrer uma segunda volta, foi mesmo aprovado por unanimidade na reunião do secretariado do PS que se realizou no domingo à tarde no Largo do Rato. Nesse encontro, José Sócrates analisou os vários cenários possíveis e deixou claro que, se houvesse segunda volta e o candidato de esquerda a passar fosse Manuel Alegre, o PS daria o seu apoio incondicional para a eleição do vice-presidente da Assembleia da República. Público 24/01/2006

3. Introduction “If people do not believe that mathematics is simple, it is only because they do not realize how complicated life is.” (John von Neumann) “We, humans, cannot survive without interacting with other humans, and ironically, it sometimes seems that we have survived despite those interactions.” (Levent Koçkesen)

4. Elements of a Game:Strategic Environment Players decision makers Strategies feasible options Payoffs objectives

5. Elements of a Game:The Rules Timingof moves Simultaneous or sequential? Nature of conflictand interaction Are players’ interests in conflict or in cooperation? Will players interact once or repeatedly? Informationalconditions Is there full information or advantages? Enforceability ofagreements orcontracts Can agreements to cooperate work?

6. Elements of a Game:Assumptions “I can calculate the motions of heavenly bodies, but not the madness of people” Isaac Newton (upon losing £20,000 in the South Sea Bubble in 1720) Rationality Players aim to maximize their payoffs Players are perfect calculators CommonKnowledge “I Know That You Know That I Know…” (popular saying)

7. Interests • Zero sum: a game in which one player's winnings equal the other player's losses • Variable-sum (non-zero sum): a game in which one player's winnings may not imply the other player's losses

8. Type of Games Static Games ofComplete Information yes Is it a one-move game? Dynamic Games ofComplete Information no yes Are all the payoffs known? Static Games ofIncomplete Information no yes Is it a one-move game? no Dynamic Games ofIncomplete Information

9. Characteristics Static Games ofComplete Information • all the payoffs are know • players simultaneously choose a strategies • the combinations of strategies may be represented in a normal-form representation How to predict the solution of a game? Player B Strategy B1 Strategy B2 Payoff 1 Payoff 2 Player A Strategy A1 Payoff 3 Payoff 4 Strategy A2

10. InvisibleHand Characteristics Static Games ofComplete Information “Every individual necessarily labours to render the annual revenue of the society as great as he can. He generally neither intends to promote the public interest, nor knows how much he is promoting it (...) By pursuing his own interest he frequently promotes that of the society more effectually than when he really intends to promote it.” Adam Smith in The Wealth of Nations Does the invisible hand exist?

11. Characteristics DominantStrategy InvisibleHand Static Games ofComplete Information Dominant Strategy: A strategy that outperforms all other choices no matter what opposing players do Dominant Strategy Equilibrium

12. InvisibleHand Prisoner’sDilemma DominantStrategy Characteristics Static Games ofComplete Information Prisoner's Dilemma Suspect B Confess Deny Suspect A Confess -36 -60 -36 -1 -3 Deny -1 -60 -3 Not a Pareto efficiency!

14. Characteristics NashEquilibrium DominantStrategies Prisoner’sDilemma InvisibleHand Static Games ofComplete Information • a player’s best decision is dependent on the other players’ decisions Nash Equilibrium: Each player chooses its best strategy according to the other players’ best strategy

15. NashEquilibrium Characteristics DominantStrategies Prisoner’sDilemma InvisibleHand Static Games ofComplete Information Company B High Low Then how to overcome several Nash equilibriums of a game? 10,10 6,4 High Company A 2,2 5,5 Low Player B Left Right 2,2 0,2 several Nash Equilibriums may coexist in the same game… Left Player A 2,0 1,1 Right

16. DominatedStrategies Characteristics DominantStrategies Prisoner’sDilemma NashEquilibrium InvisibleHand Static Games ofComplete Information Player B Left Right Middle 1,0 1,2 0,1 Top Player A 0,3 0,1 2,0 Bottom • Verify the existence of dominated strategies of one player • Re-design the normal-form representation • Verify the existence of dominated strategies of the other player • Re-design the normal-form representation • … And is it possible that a game doesn’t have a single Nash Equilibrium?

17. Characteristics DominatedStrategies DominantStrategies Prisoner’sDilemma NashEquilibrium MixedStrategies InvisibleHand Static Games ofComplete Information Mixed Strategy: A strategy in which the players judge their decision based on a degree of probability Company B High Low pA 3,6 6,2 pA= 5/7 High Company A 5,1 1,4 Low pB= 3/7 pB

18. Characteristics Dynamic Games ofComplete Information Imperfect: although the players know the payoffs, playing simultaneously disables them to have the perfect information The information can be Perfect: occurs when the players know exactly what has happened every time a decision needs to be made

19. Characteristics Dynamic Games ofComplete Information One Shot • Players don’t know much about one another • Players interact only once Repeated Indefinitely versus Finitely? • Finite • No incentive to cooperate • There's a future loss to worry about in the last period • Infinite • Cooperation may arise! • Reputation concerns matter • The game doesn’t need to be played forever, what matters is that the players don’t realize when the game is going to end

20. Characteristics SimultaneousDecision Dynamic Games ofComplete Information Simultaneous Decision (imperfect information) How to think? Must anticipate what your opponent will do right now, recognizing that your opponent is doing the same Rationality of the Players • Put yourself in your opponent’s shoes • Iterative reasoning

21. Cooperation Characteristics SimultaneousDecision Dynamic Games ofComplete Information Keep in mind If you plan to pursue an aggressive strategy ask yourself whether you are in a one-shot or in a repeated game. If it’s a repeated game: THINK AGAIN

22. Cooperation Characteristics SimultaneousDecision Dynamic Games ofComplete Information “If it’s true that we are here to help others, then what exactly are the others here for?” George Carlin Is cooperation impossible if the relationship between players is for a fixedandknown length of time? Answer: We never know when “the game” (interaction between players) will end! Struggle between high profits today and a lasting relationship into the future

23. Strategies Cooperation Characteristics SimultaneousDecision Dynamic Games ofComplete Information Tit-for-Tat Strategy • Players cooperate unless one of them fails to cooperate in some round of the game. • The others do in the next round what the uncooperative player did to them in the last round Trigger Strategy • Begin by cooperating • Cooperate as long as the rivals do • After a flaw, strategy reverts to a period of punishment of specified length in which everyone plays non-cooperatively Grim Trigger Strategy • Cooperate until a rival deviates • Once a deviation occurs, play non-cooperatively for the rest of the game

24. SequentialGames Strategies Cooperation Characteristics SimultaneousDecision Dynamic Games ofComplete Information Sequencial Decision (Perfect Information) “Loretta’s driving because I’m drinking and I’m drinking because she’s driving” in “The Lockhorns Cartoon” Games in which players make at least some of their decision at different times

25. SequentialGames Strategies Cooperation Characteristics SimultaneousDecision Payoff A1 strategy A1 Payoff B1 strategy B1 strategy A2 strategy B2 Payoff B2 Dynamic Games ofComplete Information Kuhn´s Theorem: Every game of perfect information with a finite number of nodes, has a solution to backward induction • represented in extensive form, using a game tree Corollary: If the payoffs at all terminal nodes are unequal (no ties) then the backward induction solution is unique

26. SequentialGames Strategies Cooperation Strategy Characteristics SimultaneousDecision Dynamic Games ofComplete Information Rollback or Backward Induction • Must look ahead in order to know what action to choose now • The analysis of the problem is made from the last play to the first • Look forward and reason back How to solve the game? • Start with the last move in the game • Determine what that player will do • Trim the tree • Eliminate the dominated strategies • This results in a simpler game • Repeat the procedure

27. SimultaneousDecision SequentialGames Strategies Cooperation Strategy Characteristics out 0 , 100 E in -50 , -50 fight M acc 50 , 50 Dynamic Games ofComplete Information Entrant makes the first move(must consider how monopolist will respond) If Entrant enters Monopolist accommodates

28. SequentialGames Strategies Cooperation Strategy Characteristics SimultaneousDecision Dynamic Games ofComplete Information Is there a First Mover advantage? Depends on the game! Normally there's a first move advantage: First player can influence the game by anticipation But there are exceptions! Example: Cake-cutting: one person cuts, the other gets to decide how the two pieces are allocated

29. SequentialGames Strategies Cooperation Strategy Bargaining Characteristics SimultaneousDecision Dynamic Games ofComplete Information “Necessity Never Made a Good Bargain” Benjamin Franklin • The “Bargaining Problem” arises in economic situations where there are gains from trade: • the size of the market is small • there's no obvious price standards • players move sequentially, making alternating offers • under perfect information, there is a simple rollback equilibrium • Example: when a buyer values an item more than a seller. • I value a car that I own at 1000€. If you value the same car at 1500€, there is a 500€ gain from trade (M). The question is how to divide the gains, for example, what price should be charged?

30. SequentialGames Strategies Cooperation Strategy Bargaining Characteristics SimultaneousDecision Dynamic Games ofComplete Information Take-it-or-leave-it Offers • Consider the following bargaining game for the used car: • I name a take-it-or-leave-it price • If you accept, we trade • If you reject, we walk away • Advantages • Simple to solve • Unique outcome • Disadvantages • Ignore “real” bargaining (too trivial) • Assume perfect information; we do not necessarily know each other’s values for the car • Not credible: “If you reject my offer, will I really just walk away?”

31. SequentialGames Strategies Cooperation Strategy Bargaining Characteristics SimultaneousDecision Dynamic Games ofComplete Information Who has the advantage in playing first? Depends… Value of the money in the future(discount factor) • Patience • If players are patient: • - Second mover is better off! • - Power to counteroffer is stronger than power to offer • If players are impatient • - First mover is better off! • - Power to offer is stronger than power to counteroffer

32. SequentialGames Strategies Cooperation Strategy Bargaining Characteristics SimultaneousDecision Dynamic Games ofComplete Information COMMANDMENT: In any bargaining setting, strike a deal as early as possible! Why doesn’t it happen naturally? • “Time has no meaning” • Lack of information about values! • (bargainers do not know one another’s discount factors) • Reputation-building in repeated settings! • (looks like “giving in”) Nevertheless, bargaining games could continue indefinitely… In reality they do not. Why not? • Both sides have agreed to a deadline in advance • The gains from trade, M, diminish in value over time (at a certain date M=0) • The players are impatient (time is money!)

33. SequentialGames Strategies Cooperation Strategy Bargaining Characteristics SimultaneousDecision Dynamic Games ofComplete Information Lessons Buyer: Good guy - see the seller’s points of view (“put yourself in the other’s shoes”) Seller: - create the “invisible buyer” (put pressure on the buyer) • Both: • achieve a “win-win” trade • signal that you are patient, even if you are not • For example, do not respond with counteroffers right away. Act unconcerned that time is passing-have a “poker face.” • remember that the more patient a player gets the higher fraction of the amount M that is on the table takes

34. Assumptions Static Games ofIncomplete Information Properties • at least one player is uncertain about another player’s payoff function • the importance of these analysis is related with beliefs, uncertainty and risk management Practical applications • R&D and development of products • banking and financial markets • defense - rooting terrorists

35. Assumptions Bayes’ Law Static Games ofIncomplete Information 1 .Which side of the court should I choose? 2 . The other player tries to confuse you… he moves softly to the other side Bayes’ Law is used whenever update of new information is necessary R L

36. Assumptions Bayes’ Law Revisingjudgments probability that the player 2 has a poor reception on the left probability that player 2 choose a position on the right probability that the player has a poor reception on the left giving he is on the right probability that player 2 moves to the right given that he has a poor reception on the left Static Games ofIncomplete Information Bayes’ Law:

37. Assumptions Bayes’ Law Revisingjudgments Static Games ofIncomplete Information Strategy Type Action Beliefs Payoffs Separating Strategy Pooling Strategy Strategy Spaces

38. Assumptions Bayes’ Law Revisingjudgments MixedStrategies Static Games ofIncomplete Information Mixed strategies revisited j’s mixed strategy i’s uncertainty about j’s choice of a pure strategy depends on the realization of a small amount of private information j’s choice randomization Nash equilibrium uncertainty Incomplete information

39. Assumptions • Myerson (1979) – important tool for designing games when players have private information Assumptions Bayes’ Law Revisingjudgments MixedStrategies RevelationPrinciple Examples • used in auction and bilateral-trading problems • bidder paid money to the seller and received the good • bidder must to pay an ENTRY FEE • the seller might set a RESERVATION PRICE Possibilities Static Games ofIncomplete Information • How to simplify the problem?

40. Assumptions Bayes’ Law Revisingjudgments MixedStrategies RevelationPrinciple Static Games ofIncomplete Information The seller can restrict attention to: • . The bidders simultaneously make (possibly dishonest) claims about their type (their valuations) • . For each possible combinations of claims, the sum of possibilities must be less than or equal to one 1st • TWO WAYS direct mechanism 2nd • . Restrict attention to those direct mechanisms in which it is a Bayesian Nash equilibrium for each bidder to tell the truth incentive- compatible

41. Assumptions Bayes’ Law Revisingjudgments MixedStrategies RevelationPrinciple Static Games ofIncomplete Information Theorem Any Bayesian Nash equilibrium of any Bayesian game can be represented by an incentive-compatible direct mechanism • If all other players tell the truth, then they are in effect playing the strategies • Truth-telling is an equilibrium, it is a Bayesian Nash equilibrium of the static Bayesian game

42. Characteristics Dynamic Games of Incomplete Information Revision Dynamic games:Sequential Games One player plays after the other Incomplete Information: At least one player doesn’t know the other players’ payoff. They hold Beliefs about others’ behavior – which are updated using Bayes’ Law … They may try to mislead, trick or communicate… To solve this games a new equilibrium has to be found.

43. Perfect Bayesianequilibrium Characteristics Dynamic Games of Incomplete Information Requirements At each information set the player with the move must have a belief about each node in the information set has been reached by the play of the game. Given their beliefs, the player’s strategies must be sequentially rational. Beliefs are determined by Bayes’ rule and the players’ equilibrium strategies. Belief – Probability distribution over the nodes in the information set. Sequentially rational – the action taken by the player with the move must be optimal given the player’s belief. Information set for a player – it’s a collection of decision nodes satisfying: the player has the move at every node and that he has same set of feasible actions at each node.

44. Perfect Bayesianequilibrium Characteristics SignalingGames Dynamic Games of Incomplete Information 1. Nature draws a type t for the Sender from a set of feasible types. 2. The Sender observes t and then chooses a message m from a set of feasible messages.

45. Perfect Bayesianequilibrium Characteristics SignalingGames SELL BUY BUY PASS BUY Dynamic Games of Incomplete Information 3. The Receiver observes m (but not t) and then chooses an action a from a set of feasible actions. 4. Payoffs are given to the Sender and Receiver.

46. Perfect Bayesianequilibrium Characteristics SignalingGames Dynamic Games of Incomplete Information Job Signaling Sender: worker Type: worker’s productive ability Message: worker’s education choice Receiver: market of prospective employers Action: wage paid by the market Corporate Investment Sender: firm needing capital to finance new project Type: the profitability of the firm’s existing assets Message: firm’s offer of an equity stake Receiver: potential investor Action: decision about whether to invest

47. Perfect Bayesianequilibrium Characteristics SignalingGames Perfect BayesianEquil. in SG Dynamic Games of Incomplete Information Requirements 1. After observing any message the Receiver must have a belief about which types could have sent m. ∑p(ti|mj)=1 p q … 2. For each m, the Receiver’s action must maximize the Receiver’s expected utility. The Sender’s action must maximize the Sender’s Utility. 3. The Receiver’s Belief, at any given point, follows from Bayes’ Rule.

48. Perfect Bayesianequilibrium Characteristics SignalingGames Perfect BayesianEquil. in SG CorporateInvestment Dynamic Games of Incomplete Information Corporate Investment Situation: João Silva is an entrepreneur and wants to undertake a new project in his enterprise. He has information about the profitability of the existing company, but not about the new project. He needs outside financing. Question: What will the equity stake be?

49. Characteristics Perfect Bayesianequilibrium SignalingGames CorporateInvestment Perfect BayesianEquil. in SG Dynamic Games of Incomplete Information How to turn this problem into a signaling game? Investor accepts IP= %i of the profit EP= %e of the profit João offers an equity stake s to a potential investor Investor rejects Examples IP= the investment saved in a bank EP=not giving up the company Required investment I Probability(=L)=p The investor will accept if and only if: The investor will accept if and only if: Stake offered≤ Relative return of the project His share of the expected profit≥ investment saved in a bank Separating equilibrium Pooling equilibrium The high-profit type must subsidize the low profit type. Different types offer different stakes.

50. Perfect Bayesianequilibrium Characteristics SignalingGames Perfect BayesianEquil. in SG CorporateInvestment JobSignaling Dynamic Games of Incomplete Information Job Signaling Situation: An employer wants to sort among future employees. Sender: Employees Type: Bright or Dull Msg: Beach or College Receiver: Employer Action: Hire or Reject No sender wishes to deviate from the strategy, given the Receiver’s hiring policy; Hiring is better for the Receiver given the Sender’s contingent strategy. Question: What is the perfect Bayesian Equilibrium of this game?