14.12 G am e Theory L ecture N otes Introd uction M u h a met Yild iz (Lecture 1) Gam e Theor y is a misnomer for Multiperson Decision Theory, analyzing the decision- making process when there are more than one decision-m akers where eac h agent’s pa yo ff possibly depend s on the actions taken b y the other agen ts. Since an agen t’s preferences on his actions depend on whic h actions the other parties take, his action depends on his beliefs about what the oth ers do . Of course, wh at the others do depends on their beliefs about what eac h agent does. In this w a y, a player’s action, in principle, depends on the actions available to eac h agent, eac h agent’s preferences on the outcom es, each player’s beliefs a bout w hich actions are available to eac h pla yer and h ow each player ranks th e outcomes, and further his beliefs about eac h player ’s beliefs, ad infinitum. Under perfect com petition, there are also more than one (in fact, infinitely many ) decision makers. Yet, their decision s are assum ed to be decentra lized. A consumer tries to choose the best consumption bundle that he can afford, giv en the prices — without pa ying atten tion what the other consumers do. In reality, the future prices are not know n. C onsu mers’ decisions depend on their expectations about the future prices. A n d the futu re p rices depend on con sumers’ d ecision s today. O nce again, even i n perfectly competitive enviro nment s , a consum er’s decisio ns are affected by their beliefs about what other c onsumers do — in a n aggregate level. Wh en agen ts think through what the other players w ill do, taking wh at the other players think about them into accoun t, they may findaclearwaytoplaythegame. Consid er the follow ing “game”: 1 1 \ 2L m R T (1, 1) (0, 2) (2, 1) M (2, 2) (1, 1) (0, 0) B (1, 0) (0, 0) (−1, 1) Here, Players 1 has strategies, T, M, B and Player 2 has strateg ies L, m, R. (They pick their strategies simultaneously.) The pa y offs for players 1 and 2 are indicated by the numbers in pa rent heses, the first one for player 1 an d the secon d one for player 2. For instance, if P layer 1 plays T and Player 2 pla ys R, then Player 1 gets a payo ff of 2 and Playe r 2 gets 1. L et’s assum e that each pla yer kno w s that these are the strategies and the payo ffs, each pla yer know s that eac h pla ye r kno w s this, each pla ye r kno w s that each player kn ows that each player kno w s this, ad infinitum. Now, player 1 looks at his payoffs, and realizes that, no matter what the other player plays, it is better for him to play M rather than B. Th at is, if 2 play s L, M gives 2 and B gives 1; if 2 play s m, M gives 1, B gives 0; an d if 2 pla y s R, M gives 0, B gives -1. Therefor e, he realizes that he sho uld not play B. 1 NowhecomparesTandM.Herealizes that, i f Player 2 plays L or m, M i s better than T, but if she p lay s R, T is definitely better than M . Would P la y er 2 pla y R? W hat would she pla y? To find an answer to these questions, Player 1 looks at th e g am e f rom Player 2’s point o f view . He realizes that, for P layer 2, there is no s tr ategy that is outright better th an any other strategy. For instance, R is the best strategy if 1 plays B, but oth er wise it is strictly worse t han m. Wou ld Player 2 think that Player 1 wou ld play B? We ll, she kno w s that Pla ye r 1 i s trying to maximize his expected payoff,givenbythefirst en tries as everyone know s. She m u st then de duce that P la yer 1 will no t play B. T herefore, Play er 1 concludes, she will not pla y R (as it is w orse than m in t his case). Ruling out the possibilit y that Play er 2 play s R, Player 1 looks at his pa yoffs, and sees that M is now better than T, no matter what. On the other side, Player 2 goes through similar reasoning, and concludes that 1 mu st play M , and therefore pla ys L. This kind of reasoning does not always yield suc h a c lear prediction. Imagine that y ou w ant to meet with a friend in one of two places, about which y ou both are indifferent. Unfortunately, you cannot commun icate with eac h other un til you meet. This situation 1 After all, he cannot have any belief about what Player 2 plays that would lead him to play B when M is available. 2 is forma lized in the following game, wh ich is called pure coor dination gam e: 1 \ 2Left Right Top (1,1) (0,0) Bottom (0,0) (1,1) Here, Pla ye r 1 c h ooses bet ween Top and Bottom rows, wh ile Player 2 c h ooses between Left and Right column s. In each box, the first and the second num bers denote the von Neumann-Mo rgenstern utilities of players 1 and 2, respectively. Note that Player 1 prefers Top to Bottom if he kno ws that Player 2 pla ys Left; he prefers Bottom if he knows that Player 2 plays Right. He is indifferen t if he t hinks that the other p la yer is lik ely to play either strategy with equal probab ilities. Similarly, Player 2 prefers Left if she knows that player 1 pla ys Top. There is n o c lear p rediction about the outcome o f this game. One ma y look for the stable outcomes (strategy profiles) in the sense that no player has incentiv e to deviate if he knows that the other pla y ers pla y the prescribed strategies. Here, Top-Left and Bottom-Righ t are suc h outcomes. But Bottom-Left and Top-Right are not stable in this sense. Fo r instan ce, if Bottom -L eft is kno wn to be played, eac h player would like to deviate — as it is sho wn in the following figure: 1 \ 2Left Right Top (1,1) ⇐⇓(0,0) Bottom (0,0) ⇑=⇒ (1,1 ) (Here, ⇑ means pla yer 1 deviates to Top , etc.) Un like in this gam e , m ostly players have differen t p references on the outcomes, in- ducing conflict. In the follo w ing gam e, which is known as the Battle of Sexes,conflict and the need for coordination are present together. 1 \ 2Left Right Top (2,1) (0,0) Bottom (0, 0) (1 ,2) Here, once again play ers would like to coordinate on Top-Left or Bottom-Righ t, but now Pla yer 1 prefers to coordinate on Top-L eft, wh ile Player 2 prefers to coordinate on Bottom -Right. The stable outcomes are again Top-Left and Bottom - Right . 3 2 1 2 TB L LRR (2,1) (0,0) (0,0) (1,2) Figure 1: No w, in the Battle of S exes, imagine th at Pl a y er 2 kno ws wha t Pl a y er 1 does w hen she takes her action. This can be form alized via the follow ing tree: Here, Pla ye r 1 chooses between Top a n d Bottom , t h en (knowing what Player 1 has c ho sen ) Playe r 2 ch ooses between Left a nd Right. Clearly, now Pla ye r 2 w o uld choose Left if Player 1 p lay s Top, and c hoose Right if Player 1 play s B o ttom. K nowing this, Player 1 w ould play Top. Therefore, one can argue that the only reasonable outcome of this game is To p-L eft. (This kind of reasoning is called backward indu ction.) W hen Player 2 is to ch eck wh at the other pla yer d oes, he gets only 1, while Player 1 gets 2. (In the previous game, two outcomes were stable, in whic h Pla yer 2 wou ld get 1 or 2.) That is, Player 2 prefers that P layer 1 has infor m ation about what Player 2 does, rather than she herself has inform ation about what pla yer 1 does. When it is common knowledgethataplayerhassomeinformationornot,theplayermayprefernottohave that inform a tion — a robust fact that we will see in various contexts. Exercise 1 Clearly, this is generated by the fact that Player 1 know s that Player 2 will know what Player 1 does when she moves. Consid er the situation that Player 1 thinks that Play er 2 will k n ow what P la yer 1 does o n ly with p robability π<1,andthis prob ability does not depend on what Player 1 does. What w ill happe n i n a “re asonable” equilibrium? [By the end of this course, hopefully, you will be able to formalize this 4 situatio n, and compu te the equilibr ia .] Anoth er in terp reta tion is tha t Player 1 can co m municate to Player 2, w h o cannot comm unicate to play er 1. This enables pla yer 1 to commit to his actions, pro viding a strong position in the relation. Exercise 2 Consider the following version of t he last game: after knowing what Player 2 does, Player 1 gets a chance to change his action; then, the game ends. In other words, Player 1 chooses b etween Top and Bottom ; knowing P layer 1’s choice, P layer 2 chooses between Left and Right; knowing 2’s choice, Player 1 d ecides whether to stay w here he is or to change his position. What is the “reasonable” outcome? What wou ld happen if changin g his action would cost player 1 c utiles? Imagin e that, before playin g the Battle of Sexes, Player 1 has the o p tion of e xitin g, in wh ich c a se e a ch p layer will get 3 / 2, or playing the B attle o f S exes. When asked to pla y, Player 2 will know that Pla yer 1 c ho se to play the Battle of Sexes. Ther e are two “reasonable” equilibria (or stab le outco m es). One is that Player 1 exits, thinking that, if he plays the Battle of Sexes, they will p lay the Bottom -R ight equilibrium of the Battle of Sexes, yielding only 1 for pla yer 1. The second one is that Player 1 c hooses to P lay the Battle o f Sexes, a nd in the B attle of Sexes they play To p-Left equilib rium. 2 1 Left Right Top (2,1) (0,0) Bottom (0,0) (1,2) 1 Play Exit (3/2,3/2) Some would argue that the first outcome is n ot really reasonable? Beca use, when askedtoplay,Player2willknowthatPlayer1haschosentoplaytheBattleofSexes, forgo in g the pa yoff of 3/2. She must therefore realize t hat Pla ye r 1 ca nn ot possibly be 5 planning to play Bottom, which yields the pay off of 1 max. That is, when asked to pla y, Player 2 s hould understand t hat Pla yer 1 is planning t o play Top, a nd thus she should play L eft. Ant icipating this, Player 1 s hould c hoose to pla y the Battle of Sexes game, in whic h they play Top-Left. Therefor e, the second outcome is the only reasonab le one. (This kind of reasoning is called Forwar d Induction.) Here are some more examples of gam es: 1. Prisoners’ Dilem ma: 1 \ 2 C onfess Not Confess Confess (-1, -1) (1, -10) Not Co nfess (-10, 1) (0, 0) This is a well known game that most of yo u kno w . [It is also discussed in Gibbons.] In this game no matter what the other pla yer does, eac h pla yer w ould like to confess, yielding (-1,-1), which is do m ina ted by (0,0). 2. Ha w k-D ove game 1 \ 2Hawk Dove Ha wk ¡ V −C 2 , V −C 2 ¢ (V , 0) Dov e (0,V ) ( V 2 , V 2 ) This is a generic biological game, but is also quite similar to many games in econom ics and political science. V is the v alue of a resource that one of the pla yers will enjo y. If th ey shar ed the resource, th eir values are V/2. Hawk stands for a “ tou gh” strategy, w h ereby the player does not give up the resource. H owever, if the other play er is also playing haw k, th ey end up fighting, and incur t he cost C/2 each . O n the other hand, a Hawk player gets the whole resource for itself when pla ying a Dov e. When V>C, w e ha ve a P risoners’ Dilemma game, where we would observe figh t. When w e have V<C,sothatfigh ting is costly, this game is similar to another well-k nown game, inspired by the m ovie Rebel W itho ut a C a use , named “C hicken”, where t wo players driving towa rds a cliff have to decid e wheth er to stop or continue. The one who s tops first l oses face, but may save his life. More generally, a class of games called “wars of attrition” are used to model this type of situations. In 6 this case, a player would like to play Ha wk i f his opponent play s Dove , an d play Do v e if hi s opponen t plays Ha wk. 7 14.12 G am e Theory L ecture N otes Theory of Choice M u h a met Yild iz (Lecture 2) 1 The basic theory of c hoice We consider a set X of altern atives. Alternative s are mutually exclu sive in th e sense that one cannot choose t wo distinct alternatives at the same time. We also take the set of feasible alterna tive s exhaustive so that a pla yer’s cho ices w ill always be defined. Note that this is a ma tter of modeling. For instanc e, if we have options Coffee and Tea, we define alternatives as C = Coffee but no Tea, T = Tea but no Coffee, CT = Coffee and Te a, and NT = no Co ffee and no Tea. Take a relation º on X. Note that a relation on X is a subset of X × X.Arelation º is said to be complete if and on ly i f, give n any x, y ∈ X,eitherx º y or y º x.A relation º is said to be transitive if and only if, given any x, y, z ∈ X, [x º y an d y º z] ⇒ x º z. Arelationisapreference relation if and only if it is complete and transitiv e. Giv en any preference relation º,wecandefin e strict preference  by x  y ⇐⇒ [x º y and y 6º x], and the indifference ∼ by x ∼ y ⇐⇒ [x º y and y º x]. Apreferencerelationcanberepresen ted by a ut ility functio n u : X → R in the follow in g sense: x º y ⇐⇒ u(x) ≥ u(y) ∀x, y ∈ X. 1 The follo w ing theorem states further that a relation needs to be a preference relation in order to be represented by a utility function. Theorem 1 Let X be finite. A re latio n ca n be presented by a utility functio n if and only if it is co mple te and tra n s itiv e. Moreover, if u : X → R represents º,andiff : R → R is a stric tly increasin g function, then f ◦ u also represents º. By the last statement, we call such utility functions ordin al. In order to u se th is ord inal theor y of ch oic e, w e sh ould k now the agent’s preferenc es on the alternatives. As w e have seen in the previous lecture, in game theory, a player ch ooses between his strategies, and his preferences on his strategies depend on the strategies pla yed b y the other players. Typ ically, a player does not know which strategies the other pla yers pla y. Therefore, we need a theory of decisio n-m akin g under uncertainty. 2 Decision-making under uncertainty We con sider a finite set Z of prizes, and the set P of all probability distrib u tion s p : Z → [0, 1] on Z,where P z∈Z p(z)=1. We call these prob ability distr ibu tions lotteries. A lottery can be dep icted b y a tree. For examp le, in Figure 1, Lottery 1 depicts a situation in which if head the pla yer gets $10, and if t ail, he gets $0. Lottery 1 1/2 1/2 10 0 Figure 1: Unlike th e situation w e just described, in game theory and more broadly when agen ts make th eir d ecision und er u ncertainty, we do n ot have the lotteries as in casinos where the probabilities are generated by so m e mac h ines or given. Fortuna tely, it h as been show n by Sa vage (1954) under certain conditions that a player’s beliefs can be represented by 2 a (unique) probability distribution . Usin g these probabilities, w e can represent our acts b y lotteries. We wo uld like to have a theory that constr ucts a playe r’s preferen ces on the lotteries from his preferen ces on the prizes. There are many of them. Th e most well-kno wn–a nd the most canonical and the m ost useful–on e is the theo ry of expected utility maxim iz a- tion by Von Neum a nn a nd Morgenstern. A preference relation º on P is said to be represented by a vo n Neuman n-Morge nst ern utility function u : Z → R if and only if p º q ⇐⇒ U(p) ≡ X z∈Z u(z)p(z) ≥ X z∈Z u(z)q(z) ≡ U(q) (1) for each p, q ∈ P .NotethatU : P → R represents º in ordinal sense. That is, the agent acts as if he wants t o m a x im ize t he ex pected value of u. For in stance, the expected utilit y of Lottery 1 for our agent is E(u(Lottery 1)) = 1 2 u(10) + 1 2 u(0). 1 The necessary and sufficient conditio ns for a representation as in (1) are as follow s: Axiom 1 º is comple te and transitive. This is necessary by Theorem 1, for U represents º in ordinal sense. The seco nd condition is c alled independence axiom, stating t ha t a player’s preference between t wo lotteries p and q does not chan ge if we toss a coin and giv e him a fixed lottery r if “tail” come s up. Axiom 2 For any p, q, r ∈ P ,andanya ∈ (0, 1], ap +(1− a)r  aq +(1− a)r ⇐⇒ p  q. Let p and q be the lotteries depicted in Figure 2. Then, the lotteries ap +(1− a)r and aq +(1− a)r canbedepictedasinFigure3,wherewetossacoinbetweenafixed lottery r and our lotteries p and q. Axiom 2 stipulates that the agent w ould not change his mind after the coin toss. Therefore, our axiom can be tak en as an axiom of “dynam ic consistency ” in this sense. The third condition is purely technica l, a nd called co n tin uity axiom. It states that there are no “infinitely good” or “infinitely bad” prizes. Axiom 3 For any p, q, r ∈ P ,ifp  r, then there exist a, b ∈ (0, 1) suc h that ap +(1− a)r  q  bp +(1− r)r. 1 If Z were a continuum, like R, we would compute the expected utility of p by R u(z)p(z)dz. 3 [...]... form game corresponding to this game is HH HT TH TT Head -1 ,1 -1 ,1 1 ,-1 1 ,-1 Tail 1 ,-1 -1 ,1 1 ,-1 -1 ,1 Information sets are very important! To see this, consider the following game 7 Game 2: Matching Pennies with Imperfect Information (-1 , 1) Head (1, -1 ) Tail Head 1 (1, -1 ) Head Tail 2 Tail (-1 , 1) Games 1 and 2 appear very similar but in fact they correspond to two very different situations In Game. .. 0,0 4,1 Rationalizability or Iterative elimination of strictly dominated strategies Consider the following Extended Prisoner’s Dilemma game: 1\2 confess don’t confess run away confess -5 ,-5 0 ,-6 -5 ,-1 0 don’t confess -6 ,0 -1 ,-1 0 ,-1 0 -1 0 ,-6 -1 0,0 -1 0 ,-1 0 run away In this game, no agent has any dominant strategy, but there exists a dominated strategy: “run away” is strictly dominated by “confess” (both... confess -5 ,-5 0 ,-6 don’t confess -6 ,0 -1 ,-1 “Confess” is a strictly dominant strategy for both players, therefore (“confess”, “confess”) is a dominant strategy equilibrium 1\2 confess don’t confess confess -5 ,-5 don’t confess -6 ,0 ⇑ 4 ⇐= 0 ,-6 ⇐ =-1 ,-1 ⇑ This is the only outcome, provided that each player is rational and player 2 knows that player 1 is rational Can you show this? 11 Example: (second-price... that we have two identical risk-averse agents as above, and the insurance company Insurance company is to charge the same premium 10 P for each agent, and the risk-averse agents have an option of forming a mutual fund What is the range of premiums that are acceptable to all parties? 11 14.12 Game Theory Lecture Notes Lectures 3-6 Muhamet Yildiz We will formally define the games and some solution concepts,... choose “run away,” thus she can eliminate “run away” and consider the smaller game 1\2 confess don’t confess run away confess -5 ,-5 0 ,-6 -5 ,-1 0 don’t confess -6 ,0 -1 ,-1 0 ,-1 0 13 where we have eliminated “run away” because it was strictly dominated; the column player reasons that the row player would never choose it In this smaller game, 2 has a dominant strategy which is to “confess.” That is, if 2 is rational... Head or Tail This is a game of imperfect information (That is, some of the information sets contain more than one node.) The strategies for player 1 are again Head and Tail This time player 2 has also only two strategies: Head and Tail (as he does not know what 1 has played) The normal form representation for this game will be: 1\2 Head Tail Head -1 ,1 1 ,-1 Tail -1 ,1 1 ,-1 Game 3: A Game with Nature: (5,... 1/2 2 O Right (0, -5 ) 8 Here, we toss a fair coin, where the probability of Head is 1/2 If Head comes up, Player 1 chooses between Left and Right; if Tail comes up, Player 2 chooses between Left and Right Exercise 5 What is the normal-form representation for the following game: 1 A 2 D α δ (4,4) (5,2) 1 a (1 ,-5 ) d (3,3) Can you find another extensive-form game that has the same normal-form representation?... (1 ,-5 ) d (3,3) Can you find another extensive-form game that has the same normal-form representation? [Hint: For each extensive-form game, there is only one normal-form representation (up to a renaming of the strategies), but a normal-form game typically has more than one extensive-form representation.] In many cases a player may not be able to guess exactly which strategies the other players play In order... 1 Representations of games The games can be represented in two forms: 1 The normal (strategic) form, 2 The extensive form 1.1 Normal form Definition 1 (Normal form) An n-player game is any list G = (S1 , , Sn ; u1 , , un ), where, for each i ∈ N = {1, , n}, Si is the set of all strategies that are available to player i, and ui : S1 × × Sn → R is player i’s von Neumann-Morgenstern utility... agent is risk-neutral if and only if he has a linear Von-NeumannMorgenstern utility function 7 An agent is strictly risk-averse if and only if he rejects all fair gambles: E(u(lottery 2)) < u(0) pu(x) + (1 − p)u(y) < u(px + (1 − p)y) ≡ u(0) Now, recall that a function g(·) is strictly concave if and only if we have g(λx + (1 − λ)y) > λg(x) + (1 − λ)g(y) for all λ ∈ (0, 1) Therefore, strict risk-aversion . Top-Left or Bottom-Righ t, but now Pla yer 1 prefers to coordinate on Top-L eft, wh ile Player 2 prefers to coordinate on Bottom -Right. The stable outcomes are again Top-Left and Bottom - Right. onfess Not Confess Confess (-1 , -1 ) (1, -1 0) Not Co nfess (-1 0, 1) (0, 0) This is a well known game that most of yo u kno w . [It is also discussed in Gibbons.] In this game no matter what the other. prescribed strategies. Here, Top-Left and Bottom-Righ t are suc h outcomes. But Bottom-Left and Top-Right are not stable in this sense. Fo r instan ce, if Bottom -L eft is kno wn to be played,