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Master in Engineering Policy and Management of Technology 24 th February MicroeconomyMicroeconomy João Castro Miguel Faria Sofia Taborda Cristina Carias Game Theory

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1/52 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 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

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2/52 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)

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3/52 Elements of a Game: Strategic Environment Players decision makers Payoffs objectives Strategies feasible options

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4/52 Elements of a Game: The Rules Timing of moves Simultaneous or sequential? Informational conditions Is there full information or advantages? Nature of conflict and interaction Are players’ interests in conflict or in cooperation? Will players interact once or repeatedly? Enforceability of agreements or contracts Can agreements to cooperate work?

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5/52 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 Common Knowledge “I Know That You Know That I Know…” (popular saying)

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6/52 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

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7/52 Type of Games Static Games of Complete Information Dynamic Games of Complete Information Static Games of Incomplete Information Dynamic Games of Incomplete Information Is it a one-move game? Are all the payoffs known? yes no yes no yes no

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8/52 Static Games of Complete Information all the payoffs are know players simultaneously choose a strategies the combinations of strategies may be represented in a normal-form representation Characteristics Strategy B1Strategy B2 Player B Strategy A1 Strategy A2 Player A Payoff 1Payoff 2 Payoff 3Payoff 4 How to predict the solution of a game?

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9/52 Static Games of Complete 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? Invisible Hand Characteristics

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10/52 Static Games of Complete Information Characteristics Dominant Strategy Dominant Strategy: A strategy that outperforms all other choices no matter what opposing players do Dominant Strategy Equilibrium Invisible Hand

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11/52 Static Games of Complete Information ConfessDeny Suspect B Confess Deny Suspect A Prisoner's Dilemma Characteristics Dominant Strategy Prisoner’s Dilemma Not a Pareto efficiency! Invisible Hand

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13/52 Static Games of Complete Information a player’s best decision is dependent on the other players’ decisions Characteristics Nash Equilibrium: Each player chooses its best strategy according to the other players’ best strategy Nash Equilibrium Dominant Strategies Prisoner’s Dilemma Invisible Hand

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14/52 2,2 2,01,1 0,2 Static Games of Complete Information Nash Equilibrium Characteristics Dominant Strategies Prisoner’s Dilemma Invisible Hand 10,10 2,25,5 6,4 High Low High Company A Company B Low several Nash Equilibriums may coexist in the same game… Left Right Left Right Player B Player A Then how to overcome several Nash equilibriums of a game?

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15/52 Static Games of Complete Information Dominated Strategies Characteristics Dominant Strategies Prisoner’s Dilemma Nash Equilibrium Invisible Hand 1,0 0,30,1 1,2 2,0 Player B Left Right Middle Top Bottom Player A 1.Verify the existence of dominated strategies of one player 2.Re-design the normal-form representation 3.Verify the existence of dominated strategies of the other player 4.Re-design the normal-form representation 5.… And is it possible that a game doesn’t have a single Nash Equilibrium?

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16/52 Static Games of Complete Information Characteristics Mixed Strategy: A strategy in which the players judge their decision based on a degree of probability Dominated Strategies Dominant Strategies Prisoner’s Dilemma Nash Equilibrium Mixed Strategies Invisible Hand 3,6 5,11,4 6,2 High Low High Company A Company B Low pBpB pApA p A = 5/7 p B = 3/7

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17/52 Dynamic Games of Complete Information The information can be Perfect: occurs when the players know exactly what has happened every time a decision needs to be made Imperfect: although the players know the payoffs, playing simultaneously disables them to have the perfect information Characteristics

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18/52 Players don’t know much about one another Players interact only once Dynamic Games of Complete Information Indefinitely versus Finitely? Characteristics One Shot Repeated 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

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19/52 Dynamic Games of Complete Information Simultaneous Decision (imperfect information) Must anticipate what your opponent will do right now, recognizing that your opponent is doing the same Rationality of the Players Characteristics How to think? Put yourself in your opponent’s shoes Iterative reasoning Simultaneous Decision

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20/52 Dynamic Games of Complete 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 Cooperation Characteristics Simultaneous Decision

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21/52 Dynamic Games of Complete 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 fixed and known length of time? Answer: We never know when “the game” (interaction between players) will end! Cooperation Struggle between high profits today and a lasting relationship into the future Characteristics Simultaneous Decision

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22/52 Dynamic Games of Complete 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 Strategies Cooperation 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 Characteristics Simultaneous Decision

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23/52 Dynamic Games of Complete Information Sequential Games Sequencial Decision (Perfect Information) Strategies Cooperation “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 Characteristics Simultaneous Decision

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24/52 Dynamic Games of Complete Information Corollary: If the payoffs at all terminal nodes are unequal (no ties) then the backward induction solution is unique represented in extensive form, using a game tree Sequential Games Strategies Cooperation Kuhn´s Theorem: Every game of perfect information with a finite number of nodes, has a solution to backward induction strategy A1 Payoff A1 strategy A2 strategy B1 strategy B2 Payoff B1 Payoff B2 Characteristics Simultaneous Decision

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25/52 Dynamic Games of Complete Information 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 1.Start with the last move in the game 2.Determine what that player will do 3.Trim the tree Eliminate the dominated strategies 4.This results in a simpler game 5.Repeat the procedure Sequential Games Strategies Cooperation Strategy Rollback or Backward Induction How to solve the game? Characteristics Simultaneous Decision

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26/52 Dynamic Games of Complete Information Sequential Games Strategies Cooperation Strategy E out in M fight acc 0, , , 50 Entrant makes the first move (must consider how monopolist will respond) If Entrant enters Monopolist accommodates Characteristics Simultaneous Decision

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27/52 Dynamic Games of Complete Information Sequential Games Strategies Cooperation Strategy 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 Characteristics Simultaneous Decision

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28/52 Dynamic Games of Complete Information 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? “Necessity Never Made a Good Bargain” Benjamin Franklin Sequential Games Strategies Cooperation Strategy Bargaining Characteristics Simultaneous Decision

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29/52 Dynamic Games of Complete Information 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 Take-it-or-leave-it Offers 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?” Sequential Games Strategies Cooperation Strategy Bargaining Characteristics Simultaneous Decision

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30/52 Dynamic Games of Complete 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 Sequential Games Strategies Cooperation Strategy Bargaining Characteristics Simultaneous Decision

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31/52 Dynamic Games of Complete Information “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”) 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!) COMMANDMENT: In any bargaining setting, strike a deal as early as possible! Why doesn’t it happen naturally? Nevertheless, bargaining games could continue indefinitely… In reality they do not. Why not? Sequential Games Strategies Cooperation Strategy Bargaining Characteristics Simultaneous Decision

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32/52 Dynamic Games of Complete 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 Sequential Games Strategies Cooperation Strategy Bargaining Characteristics Simultaneous Decision

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33/52 Static Games of Incomplete Information Assumptions 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

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34/52 Static Games of Incomplete Information Bayes’ Law 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 L R Assumptions

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35/52 Static Games of Incomplete Information Bayes’ Law: 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 Bayes’ Law Assumptions Revising judgments

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36/52 Static Games of Incomplete Information Strategy Action Type BeliefsPayoffs Separating Strategy Pooling Strategy Spaces Bayes’ Law Assumptions Revising judgments

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37/52 Static Games of Incomplete Information j’s mixed strategy i’s uncertainty about j’s choice of a pure strategy j’s choice depends on the realization of a small amount of private information Nash equilibrium uncertainty randomization Incomplete information Mixed strategies revisited Bayes’ Law Assumptions Revising judgments Mixed Strategies

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38/52 Static Games of Incomplete Information Assumptions Myerson (1979) – important tool for designing games when players have private information Examples used in auction and bilateral- trading problems Possibilities bidder paid money to the seller and received the good bidder must to pay an ENTRY FEE the seller might set a RESERVATION PRICE How to simplify the problem? Bayes’ Law Assumptions Revising judgments Mixed Strategies Revelation Principle

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39/52 Static Games of Incomplete Information TWO WAYS 1 st. 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 2 nd direct mechanism. Restrict attention to those direct mechanisms in which it is a Bayesian Nash equilibrium for each bidder to tell the truth The seller can restrict attention to: incentive- compatible Bayes’ Law Assumptions Revising judgments Mixed Strategies Revelation Principle

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40/52 Static Games of Incomplete Information 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 Any Bayesian Nash equilibrium of any Bayesian game can be represented by an incentive- compatible direct mechanism Bayes’ Law Assumptions Revising judgments Mixed Strategies Revelation Principle Theorem

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41/52 Dynamic Games of Incomplete Information Dynamic games: Sequential Games One player plays after the other Incomplete Information: At least one player doesn’t know the other players’ payoff. Revision 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. Characteristics

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42/52 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. Belief – Probability distribution over the nodes in the information set. Given their beliefs, the player’s strategies must be sequentially rational. Beliefs are determined by Bayes’ rule and the players’ equilibrium strategies. 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. Perfect Bayesian equilibrium Characteristics

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43/52 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. Perfect Bayesian equilibrium Characteristics Signaling Games

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44/52 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. SELL BUY PASS BUY 4. Payoffs are given to the Sender and Receiver. BUY Perfect Bayesian equilibrium Characteristics Signaling Games

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45/52 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 Perfect Bayesian equilibrium Characteristics Signaling Games

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46/52 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 q … ∑p(ti|mj)=1 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. Perfect Bayesian Equil. in SG Perfect Bayesian equilibrium Characteristics Signaling Games

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47/52 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? Perfect Bayesian Equil. in SG Perfect Bayesian equilibrium Characteristics Signaling Games Corporate Investment

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48/52 Dynamic Games of Incomplete Information How to turn this problem into a signaling game? Probability(=L)=p João offers an equity stake s to a potential investor Investor accepts Investor rejects IP= %i of the profit EP= %e of the profit IP= the investment saved in a bank EP=not giving up the company Required investment I The investor will accept if and only if: His share of the expected profit≥ investment saved in a bank The investor will accept if and only if: Stake offered≤ Relative return of the project Pooling equilibrium Separating equilibrium The high-profit type must subsidize the low profit type. Examples Different types offer different stakes. Perfect Bayesian Equil. in SG Perfect Bayesian equilibrium Characteristics Signaling Games Corporate Investment

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49/52 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 Question: What is the perfect Bayesian Equilibrium of this game? 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. Perfect Bayesian Equil. in SG Perfect Bayesian equilibrium Characteristics Signaling Games Corporate Investment Job Signaling

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50/52 Summary Static Games of Complete Information Dynamic Games of Complete Information Static Games of Incomplete Information Dynamic Games of Incomplete Information Is it a one-move game? Are all the payoffs known? yes no yes no yes no

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51/52 Recommended Readings 1.MATA, José (2000) - Economia da Empresa, Fundação Calouste Gulbenkian (http://josemata.org/ee) 2.GIBBONS, Robert (1992) - A primer in game theory, Harvester Wheatsheaf, New York 3.VARIANT, H. (2003) - Intermediate Microeconomics: A modern approach, 6th ed., W.W. Norton & Co. 4.HARSANYI, John C. (1994) - Games with incomplete information, Nobel Lecture, December 9, 1994.

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52/52 References MATA, José (2000) - Economia da Empresa, Fundação Calouste Gulbenkian GIBBONS, Robert (1992) - A primer in game theory, Harvester Wheatsheaf, New York HARSANYI, John C. (1994) - Games with incomplete information, Nobel Lecture, December 9, VARIANT, H. (2003) - Intermediate Microeconomics: A modern approach, 6th ed., W.W. Norton & Co. Economics Department, Princeton University Princeton Economic Theory Papers - The Bank of Sweden Prize in Economic Sciences in Memory of Alfred Nobel Game theory - The Center for Game Theory in Economics - The Game Theory Society - The International Society Of Dynamic Games -

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Master in Engineering Policy and Management of Technology 24 th February MicroeconomyMicroeconomy João Castro Miguel Faria Sofia Taborda Cristina Carias Game Theory Discussion

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