3 Game Theory: Basi Conepts Eah disipline of the soial sienes rules omfortably ithin its on hosen domain: : : so long as it stays largely oblivious of the others. Edard O. Wilson (1998):191 3.1 and and eat oonuts, hih dangle from a lofty branh of a palm tree. Their favorite oonut palm produes only one fruit per tree. To get the oonut, at least one of them must limb the tree and knok the oonut loose so that it falls to the ground. Careful energy measurements sho that a oonut is orth 10 K (kiloalories) of energy, the ost of running up the tree, shaking the oonut loose, and running bak don to the ground osts 2 K for, but is negligible for, ho is muh smaller. Moreover, if both individuals limb the tree, shake the oonut loose, then limb don the tree and eat the oonut, gets 7 K and gets only 3 K, beause hogs most of it; if only limbs the tree, hile aits on the ground for the oonut to fall, gets 6 K and gets 4 K ( eats some before gets bak don from the tree); if only limbs the tree, gets 9 K and gets 1 K (most of the food is gone by the time gets there). What ill and do if eah ants to maximize net energy gain? There is one ruial issue that must be resolved: ho deides first hat to do, or? There are three possibilities: (a) deides first; (b) deides first; () both individuals deide simultaneously. We ill go through the three ases in turn. Assuming deides first, e get the situation depited in Fig. 3.1. We all a figure like this a game tree, and e all the game it defines an extensive form game. At the top of the game tree is the root node (the little dot labeled ) ith to branhes, labeled (ait) and (limb). 32
Game Theory 33 This means gets to hoose and an go either left () or right (). This brings us to the to nodes labeled, in eah of hih Little John an ait () or limb (). 0,0 9,1 4,4 5,3 Figure 3.1. and : hooses first. While has only to strategies, atually has four: a. Climb no matter hat does (). b. Wait no matter hat does ().. Do the same thing does (). d. Do the opposite of hat does (). The first letter in parenthesis indiates s move if aits, and the seond is s move if limbs. We all a move taken by a player at a node an ation, and e all a series of ations that fully define the behavior of a player a strategy atually a pure strategy, in ontrast to mixed and behavioral strategies, hih e ill disuss later, that involve randomizing. Thus, has to strategies, eah of hih is simply an ation, hile has four strategies, eah of hih is to ations one to be used hen goes left, and one hen goes right. At the bottom of the game tree are four nodes, hih e variously all leaf or terminal nodes. At eah terminal node is the payoff to the to players, (player 1) first and (player 2) seond, if they hoose the strategies that take them to that partiular leaf. You should hek that the payoffs orrespond to our desription above. For instane, at the leftmost leaf hen both ait, ith neither John expending or ingesting energy, the payoff is (0,0). At the rightmost leaf both limb the tree, osting 2 K, after hih gets 7 K and gets 3 K. Their net payoffs are thus (5,3). And similarly for the other to leaves.
34 Chapter 3 Ho should deide hat to do? Clearly, should figure out ho ill reat to eah of s to hoies, and. If hooses, then ill hoose, beause this pays 1 K as opposed to 0 K. Thus, gets 9 K by moving left. If hooses, ill hoose, beause this pays 4 K as opposed to 3 K for hoosing. Thus gets 4 K for hoosing, as opposed to 9 K for hoosing. We no have ansered s problem: hoose. What about? Clearly, must hoose on the left node, but hat should he hoose on the right node? Of ourse it doesn t really matter, beause ill never be at the right node. Hoever, e must speify not only hat a player does along the path of play (in this ase the left branh of the tree), but at all possible nodes on the game tree. This is beause e an only say for sure that is hoosing a best response to if e kno hat does, and onversely. If makes a rong hoie at the right node, in some games (though not this one) ould do better by playing. In short, must hoose one of the four strategies listed above. Clearly, should hoose (do the opposite of ), beause this maximizes s payoff no matter hat does. Conlusion: the only reasonable solution to this game is for to ait on the ground, and to do the opposite of hat does. Their payoffs are (9,1). We all this a Nash equilibrium (named after John Nash, ho invented the onept in about 1950). A Nash equilibrium in a to-player game is a pair of strategies, eah of hih is a best response to the other; i.e., eah gives the player using it the highest possible payoff, given the other player s strategy. There is another ay to depit this game, alled its strategi form or normal form. It is ommon to use both representations and to sith bak and forth beteen them, aording to onveniene. The normal form orresponding to Fig. 3.1 is in Fig. 3.2. In this example e array strategies of player 1 () in ros, and the strategies of player 2 () in olumns. Eah entry in the resulting matrix represents the payoffs to the to players if they hoose the orresponding strategies. We find a Nash equilibrium from the normal form of the game by trying to pik out a ro and a olumn suh that the payoff to their intersetion is the highest possible for player 1 don the olumn, and the highest possible for player 2 aross the ro (there may be more than one suh pair). Note that
Game Theory 35 Big John 9,1 5,3 9,1 0,0 4,4 5,3 0,0 4,4 Figure 3.2. Normal form of and hen moves first..; / is indeed a Nash equilibrium of the normal form game, beause 9 is better than 4 for don the olumn, and 1 is the best Little John an do aross the ro. Can e find any other Nash equilibria to this game? Clearly.; / is also a Nash equilibrium, beause is a best reply to and onversely. But the.; / equilibrium has the drabak that if should happen to make a mistake and play, gets only 3, hereas ith, Little John gets 4. We say is eakly dominated by, meaning that pays off at least as ell for no matter hat does, but for at least one move of, has a higher payoff than for ( 4.1). But hat if plays? Then should play, and it is lear that is a best response to. So this gives us another Nash equilibrium,.; /, in hih does muh better, getting 4 instead of 1, and does muh orse, getting 4 instead of 9. Why did e not see this Nash equilibrium in our analysis of the extensive form game? The reason is that.; / involves making an inredible threat (see 4.2 for a further analysis of s inredible threat). I don t are hat you do, says I m aiting here on the ground no matter hat. The threat is inredible beause knos that if he plays, then hen it is s turn to arry out the threat to play, ill not in fat do so, simply beause 1 is better than 0. 1 We say a Nash equilibrium of an extensive form game is subgame perfet if, at any point in the game tree, the play ditated by the Nash equilibrium remains a Nash equilibrium of the subgame. The strategy.; / is not subgame perfet beause in the subgame beginning ith s hoie of on the left of Fig. 3.1 is not a best response. Nie try, anyay,! 1 This argument fails if the individuals an ondition his behavior in one day on their behavior in previous days (see hapter 9). We assume the players annot do this.
36 Chapter 3 But hat if gets to hoose first? Perhaps no an fore a better split than getting 1 ompared to s 9. This is the extensive form game (Fig. 3.3). We no all player 1 and Big John player 2. No has four strategies (the strategies that belonged to in the previous version of the game) and only has to (the ones that belonged to before). noties that s best response to is, and s best response to is. Beause gets 4 in the first ase and only 1 in the seond, Little John hooses. s best hoie is then, and the payoffs are (4,4). Note that by going first, is able to preommit to a strategy that is an inredible threat hen going seond. 0,0 4,4 1,9 3,5 Figure 3.3. and : hooses first. The normal form for the ase hen goes first is illustrated in Fig. 3.4. Again e find the to Nash equilibria (; ) and (; ), and again e find another Nash equilibrium not evident at first glane from the game tree: no it is ho has an inredible threat, by playing, to hih s best response is. The final possibility is that the players hoose simultaneously or, equivalently, eah player hooses an ation ithout seeing hat the other player hooses. In this ase, eah player has to options: limb the tree (), or ait on the ground (). We then get the situation in Fig. 3.5. Note the ne element in the game tree: the dotted line onneting the to plaes here hooses. This is alled an information set. Roughly speaking, an information set is a set of nodes at hih (a) the same player hooses, and (b) the player hoosing does not kno hih partiular node represents the atual hoie node. Note also that e ould just as ell interhange Big John and in the diagram, reversing their payoffs at the terminal nodes, of ourse. This illustrates an important point: there may be more than one extensive form game representing the same real strategi situation.
Game Theory 37 4,4 3,5 4,4 0,0 1,9 3,5 0,0 1,9 Figure 3.4. Normal form of and game hen moves first. Even though there are feer strategies in this game, it is hard to see hat an equilibrium might be by looking at the game tree. This is beause hat does annot depend on hih hoie makes, beause does not see s hoie. So let s look at the normal form game, in Fig. 3.6. From this figure, it is easy to see that both.; / and.; / are Nash equilibria, the first obviously favoring and the seond favoring. In fat, there is a third Nash equilibrium that is more diffiult to pik out. In this equilibrium randomizes by hoosing and ith probability 1=2, and does the same. This is alled a mixed strategy Nash equilibrium; you ill learn ho to find and analyze it in 3.7. In this equilibrium has payoff 4:5 and has payoff 2. The reason for this meager total payoff is that ith probability 1=4, both ait and get zero reard, and sometimes both limb the tree! 0,0 9,1 4,4 5,3 Figure 3.5. and hoose simultaneously. 5,3 4,4 9,1 0,0 Figure 3.6. and : normal form in the simultaneous move ase.
38 Chapter 3 3.2 The Extensive Form An extensive form game G onsists of a number of players, a game tree, and a set of payoffs. A game tree onsists of a number of nodes onneted by branhes. Eah branh onnets a head node to a distint tail node If b is a branh of the game tree, e denote the head node of b by b h, and the tail node of b by b t. A path from node a to node a 0 in the game tree is a sequene of branhes starting at a and ending at a 0. 2 If there is a path from node a to a 0, e say a is an anestor of a 0, and a 0 is a suessor to a. We all k the length of the path. If a path from a to a 0 has length one, e all a the parent of a 0, and a 0 is a hild of a. We require that the game tree have a unique node r, alled the root node, that has no parent, and a set T of nodes alled terminal nodes or leaf nodes, that have no hildren. We assoiate ith eah terminal node t 2 T (2 means is an element of ), and eah player i, a payoff i.t/ 2 R (R is the set of real numbers). We say the game is finite if it has a finite number of nodes. We assume all games are finite, unless otherise stated. We also require that the graph of G have the folloing tree property. There must be exatly one path from the root node to any given terminal node in the game tree. Equivalently, every node exept the root node has exatly one parent. Players relate to the game tree as follos. Eah nonterminal node is assigned to a player ho moves at that node. Eah branh b ith head node b h node represents a partiular ation that the player assigned to b h that node an take there, and hene determines either a terminal node or the next point of play in the game the partiular hild node b t to be visited next. 3 If a stohasti event ours at a node a (for instane, the eather is Good or Bad, or your partner is Nie or Nasty), e assign the fititious player Nature to that node, the ations Nature takes representing the possible outomes of the stohasti event, and e attah a probability to eah branh of hih a is the head node, representing the probability that Nature hooses that branh (e assume all suh probabilities are stritly positive). 2 Tehnially, a path is a sequene b 1 ; : : : ; b k of branhes suh that b1 h D a, bt i D bh ic1 for i D 1; : : :k 1, and b t k D a0 ; i.e., the path starts at a, the tail of eah branh is the head of the next branh, and the path ends at a 0. 3 Thus if p D.b 1 ; : : : ; b k / is a path from a to a 0, then starting from a, if the ations assoiated ith the b j are taken by the various players, the game moves to a 0.
Game Theory 39 The tree property thus means that there is a unique sequene of moves by the players (inluding Nature) leading from the root node to any speifi node of the game tree, and for any to nodes, there is at most one sequene of player moves leading from the first to the seond. A player may kno the exat node in the game tree hen it is his turn to move (e.g., the first to ases in and, above), but he may kno only that he is at one of several possible nodes. This is the situation faes in the simultaneous hoie ase (Fig. 3.6). We all suh a olletion of nodes an information set. For a set of nodes to form an information set, the same player must be assigned to move at eah of the nodes in the set and have the same array of possible ations at eah node. We also require that if to nodes a and a 0 are in the same information set for a player, the moves that player made up to a and a 0 must be the same. This riterion is alled perfet reall, beause if a player never forgets his moves, he annot make to different hoies that subsequently land him in the same information set. 4 Suppose eah player i D 1; : : : ; n hooses strategy s i. We all s D.s 1 ; : : : ; s n / a strategy profile for the game, and e define the payoff to player i, given strategy profile s, as follos. If there are no moves by Nature, then s determines a unique path through the game tree, and hene a unique terminal node t 2 T. The payoff i.s/ to player i under strategy profile s is then defined to be simply i.t/. Suppose there are moves by Nature, by hih e mean that a one or more nodes in the game tree, there is a lottery over the various branhes emanating from that node, rather than a player hoosing at that node. For every terminal node t 2 T, there is a unique path p t in the game tree from the root node to t. We say p t is ompatible ith strategy profile s if, for every branh b on p t, if player i moves at b h (the head node of b), then s i hooses ation b at b h. If p t is not ompatible ith s, e rite p.s; t/ D 0. If p t is ompatible ith s, e define p.s; t/ to be the produt of all the probabilities assoiated ith the nodes of p t at hih Nature moves along p t, or 1 if Nature makes no moves along p t. We no define the payoff to 4 Another ay to desribe perfet reall is to note that the information sets N i for player i are the nodes of a graph in hih the hildren of an information set 2 N i are the 0 2 N i that an be reahed by one move of player i, plus some ombination of moves of the other players and Nature. Perfet reall means that this graph has the tree property.
40 Chapter 3 player i as i.s/ D X t2t p.s; t/ i.t/: (3.1) Note that this is the expeted payoff to player i given strategy profile s, assuming that Nature s hoies are independent, so that p.s; t/ is just the probability that path p t is folloed, given strategy profile s. We generally assume in game theory that players attempt to maximize their expeted payoffs, as defined in (3.1). N B S Alie 0.6 0.4 Alie Bob 4,4 L R Bob 0,5 5,0 1,1 6,4 L Bob R d u d u d u u Bob d 0,0 0,0 4,6 Figure 3.7. Evaluating Payoffs hen Nature Moves For example, onsider the game depited in Figure 3.7. Here, Nature moves first, and ith probability p l D 0:6 goes B here the game beteen Alie and Bob is knon as the Prisoner s Dilemma ( 3.11), and ith probability p l D 0:4 goes S, here the game beteen Alie and Bob is knon as the Battle of the Sexes ( 3.9). Note that Alie knos Nature s move, beause she has separate information sets on the to branhes here Nature moves, but Bob does not, beause hen he moves, he does not kno hether he is on the left or right hand branh. If e rite A.x; y; z/ and B.x; y; z/ for the payoffs to Alie and Bob, respetively, hen Alie plays x 2 fl; Rg, Bob plays y 2 fu; dg, and Nature plays z 2 fb; Sg, then (3.1) gives, for instane A.L; u/ D p u A.L; u; B/ C p r A.L; u; S/ D 0:6.4/ C 0:4.6/ D 4:8I B.L; u/ D p u B.L; u; B/ C p r B.L; u; S/ D 0:6.4/ C 0:4.4/ D 4:0I A.R; u/ D p u A.R; u; B/ C p r A.R; u; S/ D 0:6.5/ C 0:4.0/ D 3:0I B.R; u/ D p u B.R; u; B/ C p r B.R; u; S/ D 0:6.0/ C 0:4.0/ D 0I The reader should fill in the payoffs at the remaining nodes.
Game Theory 41 3.3 The Normal Form The strategi form or normal form game onsists of a number of players, a set of strategies for eah of the players, and a payoff funtion that assoiates a payoff to eah player ith a hoie of strategies by eah player. More formally, an n-player normal form game onsists of a. A set of players i D 1; : : : ; n. b. A set S i of strategies for player i D 1; : : : ; n. We all s D.s 1 ; : : : ; s n /, here s i 2 S i for i D 1; : : : ; n, a strategy profile for the game. 5. A funtion i W S! R for player i D 1; : : : ; n, here S is the set of strategy profiles, so i.s/ is player i s payoff hen strategy profile s is hosen. To extensive form games are said to be equivalent if they orrespond to the same normal form game, exept perhaps for the labeling of the ations and the naming of the players. But given an extensive form game, ho exatly do e form the orresponding normal form game? First, the players in the normal form are the same as the players in the extensive form. Seond, for eah player i, let S i be the set of strategies of that player, eah strategy onsisting of a hoie of an ation at eah information set here i moves. Finally, the payoff funtions are given by equation (3.1). If there are only to players and a finite number of strategies, e an rite the payoff funtion in the form of a matrix, as in Fig. 3.2. As an exerise, you should ork out the normal form matrix for the game depited in Figure 3.7. 3.4 Mixed Strategies Suppose a player has pure strategies s 1 ; : : : ; s k in a normal form game. A mixed strategy for the player is a probability distribution over s 1 ; : : : ; s k ; i.e., a mixed strategy has the form D p 1 s 1 C : : : C p k s k ; here p 1 ; : : : p k are all nonnegative and P n 1 p j D1. By this e mean that the player hooses s j ith probability p j, for j D1; : : : ; k. We all p j the eight of s j in. If all the p j s are zero exept one, say p l D1, e say 5 Tehnially, these are pure strategies, beause later e ill onsider mixed strategies that are probabilisti ombinations of pure strategies.