ABiefIntoductiontotheBasicsof Game Theoy Roy Smead Notheasten Univesity Septembe 18, 2013 Game theoy is a set of mathematical tools that ae used to epesent and study social inteactions. This is a vey bief (and not compehensive) intoduction to some of the basic ideas of game theoy. In many ways, game theoy is an extension of decision theoy which is the study of ational choice when we ae faced with uncetainty. Fo this eason, some of the basic concepts fom decision theoy (e.g. pobability and utility) ae impotant fo undestanding the basics of game theoy. The fist section coves some cental ideas fom decision theoy and the discussion of game theoy begins in section two. 1 Decision Theoy What is the best decision to make when we ae faced with uncetainty about the outcomes? This is the cental question of decision theoy. A decision poblem typically consists of (i) a set of possible states of the wold, (ii) the pobability that each state will occu, (iii) a set of available actions to take and (iv) a value assigned to each act in each state of the wold. 1.1 Pobability A pobability of a state is just a numbe between 0 and 1 that epesents the chance that state will occu. If S is a possible event, P(S) isthepobabilitythat S will occu. In decision theoy, pobabilities ae usually intepeted as subjective degees of confidence that an event will occu. Pobability is govened by a small set of ules (called the Kolmogoov axioms): 1. All pobabilities ae in the [0, 1] inteval: 0 apple P(S) apple 1 2. The pobability of a necessay event N is equal to 1: P(N) =1 1
3. The pobability that one of two mutually exclusive events S and R occus is equal to the sum of thei pobabilities: P(S or) = P(S) + P(R) These thee axioms plus a definition fo conditional pobability P(S given R) = P(S and R)/P(R) aesu cient to define all of elementay pobability theoy. Hee ae a few facts of pobability that will be helpful fo ou puposes: S and R ae independent of one anothe if and only if P(S) =P(S given R). If S and R ae independent, P(S and R) =P(S) P(R), othewise P(S and R) =P(S) P(R given S). If S and R ae not mutually exclusive, then P(S o R) =P(S)+P(R) P(S and R). 1.2 Utility Taking a paticula action in a paticula state usually has some consequences. The desiability of those consequences is epesented by anothe numbe called utility. The utility of an outcome can be any eal numbe (positive o negative) and the highe numbe epesents moe desiability. Someone s utility is just a numeical epesentation of thei pefeences, whateve those may be. In othe wods, utility epesents what someone caes about and the degee to which they cae about it. u(x) epesentstheutilityofoutcomex. The utility of a paticula decision acoss all possible states can be epesented in a decision matix. Suppose thee ae two possible actions one might take (A and B) andtheepossibleelevantstatesofthewoldthatmightoccu(s, R, and T ). It is typical to assume that these states ae mutually exclusive and exhaustive, meaning one and only one will occu. We can epesent the decision poblem as follows: Actions States of the Wold S R T A 1 2 3 B 5 0 0 This means that the most desiable outcome is to choose act B in state S followed by act A in states T, R, ands espectively. Howeve, we can only choose ou actions and cannot choose which state of the wold will occu. So, if you want to 2
know what act is the best to do, you need to know the pobability that each state will occu. 1.3 Expected Utility We can combine the pobability that each state will occu with the utilities of choosing an action fo each state. This gives us the expected utility fo each action. Lets take the example above and suppose that the pobabilities fo each state ae as follows: P(S) =0.4, P(R) =0.4 andp(s) =0.2. Given the fo each act-state combination povided above we can calculate the expected utility of each action by combining the pobabilities with the utilities: EU(A) =P(S) u(a in S)+P(R) u(a in R)+P(T ) u(a in T ) =0.4 1+0.4 2+0.2 3=1.8 EU(B) =P(S) u(b in S)+P(R) u(b in R)+P(T ) u(b in T ) =0.4 5+0.4 0+0.2 0=2.0 Since utility epesents what one caes about (and the degee that they cae about it), one commonly accepted pinciple of ational decision making is to choose whicheve action geneates the highest expected utility. In this case, action B has highe expected utility than action A. The idea in this example can be extended and expessed vey geneally. If some action A has many possible consequences C 1,C 2,...,C n, then to figue out the expected utility fo that action, you simply conside the utility of each possible consequence multiplied by the pobability of that consequence. The expected utility is the sum of those poducts: EU(A) = X i u(c i )P(C i ). 1.4 Dominance In some special cases, one action may be bette than othe actions no matte what state of the wold we wind up in. When an one act is bette than anothe no matte what happens, this action is said to be dominant ove the othe. Thee ae two foms of dominance: weak and stong. An act stongly dominates anothe if and only if it leads to highe utility no matte what happens. An act weakly dominates anothe if and only if it does at least as good as the othe no matte what happens and does bette in at least one instance. Conside the following example: 3
Actions States of the Wold S R T A 5 1 3 B 2 1 3 C 4 0 2 In this case, we can see that act A does bette than act C no matte if we end up in state S, R o T. So, act A stictly dominates act C. Also, act A weakly dominates act B since it does bette when S occus and equally well when R o T occus. Recognizing that one action dominates anothe can be helpful if we don t know the pobabilities of the states. If an act dominates anothe, it doesn t matte what the pobabilities of the states ae, we will do at least as well o bette by choosing the dominate action. In the example hee, it would be iational to choose any action othe than A since it dominates both othe actions. 2 Game Theoy A game is essentially a multi-peson decision poblem whee the outcome depends, not on some extenal state of the wold, but on the actions of all the playes. A game consists of a set of playes, a set of stategies (actions) fo each playe and apayo function fo each playe which specifies the utility that playe i gets fo evey possible combination of stategy choices by the playes. The most well known game is pobably the Pisone s Dilemma (shown below). Playe 2 Playe 1 Coopeate Defect Coopeate 2,2 0,3 Defect 3,0 1,1 Pisone s Dilemma In this game thee ae two playes {Playe 1, Playe 2}. Playe 1 s stategy set is {Coopeate, Defect} and Playe 2 s stategy set is {Coopeate, Defect}. The payo s fo evey combination of stategies is given in the game matix above. The left numbe in each cell is the payo to Playe 1, the ight numbe in each cell is the payo to Playe 2. Hee ae some othe impotant tems and ideas fo studying games: 4
When a game is expessed in a matix-fom as above it is called then Nomal Fom o the Stategic Fom of a game. An outcome of a game can be specified by a list of eveyones stategy choices: (Playe 1 s choice, Playe 2 s choice). Such a list is called a Stategy Pofile. A best esponse to othe playes stategies is a stategy that geneates the highest payo given what the othe playes ae doing. Most games we will conside will have just two playes, like the Pisone s Dilemma. And, in many cases, the playes playes will have the same stategies available to them and have symmetic payo functions. These games ae called two-playe symmetic games. It is possible to have moe than two playes and fo di eent playes to have di eent stategies available to them. We will conside these moe complicated games late on, but fist we will focus on two-playe symmetic games to intoduce the cental solution concept fo game theoy: the Nash equilibium. 3 Nash Equilibium If we examine the Pisone s Dilemma game, we will notice that Playe 1 does bette by playing Defect no matte what Playe 2 does (i.e. if Playe 2 coopeates, Playe 1 gets 3 by playing Defect and 2 by playing Coopeate and if Playe 2 defects, Playe 1 gets 1 by playing Defect and 0 by playing Coopeate ). And likewise, Playe 2 does bette by playing Defect no matte what Playe 1 does. In othe wods, the stategy Defect stictly dominates the stategy Coopeate fo both playes. Since choosing dominant actions ove non-dominant actions is consideed the ational choice, the ational solution to the Pisone s Dilemma is fo both playes to choose Defect. In othe wods the stategy pofile (Defect, Defect) is the unique ational solution to the Pisone s Dilemma. Playe 2 Playe 1 Right Left Right 1,1 0,0 Left 0,0 1,1 The Diving Game In many games, howeve, easoning fom dominance does not poduce a solution. Conside the Diving game (above). In this game each playe has two 5
stategies: Right and Left. But, neithe stategy dominates the othe since the best thing to do depends on what the othe playe does. If the Playe 2 dives on the Right, Playe one wants to dive on the Right. And, if Playe 2 dives on the Left, Playe 1 wants to dive on the Left. So, neithe stategy dominates the othe. The idea of a Nash equilibium will allow us to find the solutions to games whee the idea of dominance doesn t help. Playes ae at a Nash equilibium of agameeachplaye sstategyisabestesponsetothestategiesofalltheothe playes. This idea can be captued as follows: Nash Equilibium: A stategy pofile is a Nash equilibium if and only if no playe could do bette by choosing a di eent stategy given what the othe playes ae doing. In a two playe game, a stategy pofile (s, ) isapue stategy Nash equilibium i u 1 (s, ) u i (s 0,) fo all s 0 in Playe 1 s stategy set AND u 2 (s, ) u 2 (s, 0 )foall 0 in Playe 2 s stategy set. In the Pisone s Dilemma, thee is only one Nash equilibium: (Defect, Defect). In the Diving Game, thee ae at least two Nash equilibia: (Right, Right) and (Left, Left) in eithe of those states, neithe playe wants to change given what the othe playe is doing. Thee is a thid Nash equilibium in the Diving Game as well, but this thid equilibium is not a pue stategy equilibium and involves what ae called mixed stategies (see the next section). The idea of a Nash equilibium also woks in asymmetic games, when playes might have di eent stategies and di eent payo functions. Conside the game below whee the two playes have di eent stategies and the payo functions ae also not symmetic. Playe 2 Playe 1 a b c d 2,1 0,2 2,5 1,0 s 4,0 2,1 1,1 1,1 t 3,3 2,0 0,0 1,3 Example asymmetic game To find all the pue stategy Nash equilibia of this game, we simply have to go though evey possible combination of stategies and ask whethe eithe playe would want to switch given what the othe is doing. Doing an exhaustive seach on 6
the example game above shows us that thee ae fou pue stategy Nash equilibia in this game: (, c), (s, b), (s, d) and(t, d). Notice that thee of these equilibia actually involve (weakly) dominated stategies. Fo Playe 1, s weakly dominates t since it does bette against a and c and equally well against b and d. Likewise, fo Playe 2, c weakly dominates b. Despite this, howeve, these stategies can be pat of Nash equilibia of the game. Nash equilibia can involve stategies that ae weakly dominated, but not stategies that ae stongly dominated. So a Nash equilibium captues a solution to the game. But what does that mean? It tuns out that thee is not a clea consensus on what the solution is supposed to tell us. Does it tell us how we ought to play the game? Does it make apedictionabouthowpeoplewill play the game? O does it just tell us what idealized (and fictitious) ational beings would do if they played a game and if that is all it tells us, why is that useful to know? It tuns out that the concept of a Nash equilibium is useful no matte what you think the solution eally means. It tells us what we ought to do if we want to get the most we can out of agameifeveyoneelseistyingtodothesame(andweallknowthatweaeall tying to do this, and we all know that we all know, etc). But, the Nash equilibia of a game ae also the points that will goven the way that individuals will lean when playing a game. They also tell us something about how mindless agents (like bacteia) might evolve when playing a game. These ae topics that will be discussed late on. One inteesting fact abut all finite games games with a finite numbe of stategies and a finite numbe of playes is that all games have at least one Nash equilibium. This means that all games have at least one ational solution. If the games ae lage and complicated enough, it may be too di cult to find the solution even though we know thee is one. And in some games, the only Nash equilibia cannot be expessed in tems of a specific set of stategies and involve playes mixing between stategies of the game. 4 Mixed Stategies Conside the well known game Rock-Pape-Scissos (expessed in nomal fom below). We can wok though evey possible combination of stategies to pove what we all aleady know: no matte combination of stategies you choose, someone wants to switch to a di eent stategy. This means that thee is no pue stategy Nash equilibium in this game. Instead, the best thing to do in this game is to be unpedictable. We might ty to choose a stategy at andom. Simply playing in Rock- Pape-Scissos is called a pue stategy but when we ty to act unpedicatbly 7
Playe 2 Playe 1 p s 0,0-1,1 1,-1 p 1,-1 0,0-1,1 s -1,1 1,-1 0,0 Rock-Pape-Scissos sometimes using one stategy, sometimes anothe we ae using a mixed stategy. A mixed stategy is defined by giving a pobability to each of the pue stategies in the game. We use the symbol to epesent a mixed stategy and s to epesent pue stategies. A mixed stategy specifies the chance that a playe chooses each pue stategy. Fo example, the mixed stategy whee you choose, p and s with equal chance is: =(1/3, 1/3, 1/3). And the mixed stategy whee we play half the time and p and s one quate of the time each is: =(1/2, 1/4, 1/4). We can also have mixed stategies that ignoe some of the pue stategies. Fo example, the mixed stategy that only uses and s with equal chance is: =(1/2, 0, 1/2). When you know opponent is using a mixed stategy, you can conside the expected utility fo each action you might take. Suppose Playe 1 believes Playe 2willuseamixedstategy 2 =(1/2, 2/6, 1/6) in Rock-Pape-Scissos. If P 2 (s) is the pobability that Playe 2 uses stategy s, then Playe 1 can calculate how well she expects each of he stategies to do against Playe 2 using the idea of expected utility. Fo example, hee is Playe 1 s expected utility fom playing : EU(, 2) =P 2 ()u 1 (, )+P 2 (p)u 1 (, p)+p 2 (s)u 1 (, s) EU(, 2) =(1/2)(0) + (2/6)( 1) + (1/6)(1) = 1/6 Using the same method to calculate his expected utility fo he othe two stategies shows that playing p would give the best expected esult: EU(p, 2) =(1/2)(1) + (2/6)(0) + (1/6)( 1) = 2/6 EU(s, 2) =(1/2)( 1) + (2/6)(1) + (1/6)(0) = 1/6 So, if Playe 1 thinks that Playe 2 is going to use this stategy, she should just play p and she will win moe often than she loses. 8
Howeve, if Playe 2 can anticipate that Playe 1 might figue out how to exploit any tendencies towad one stategy ove anothe, Playe 2 might decide to just play all stategies with equal chance. And, if Playe 2 uses the mixed stategy =(1/3, 1/3, 1/3), Playe 1 will not be able to take advantage of any tendency. In othe wods, if Playe 2 uses the =(1/3, 1/3, 1/3) stategy, Playe 1 will be indi eent to all he stategies. Futhemoe, if Playe 1 easons the same way and is woied about Playe 2 potentially taking advantage of any tendencies, Playe 1 will also use the = (1/3, 1/3, 1/3) stategy. When both playes use this stategy in Rock-Pape- Scissos, neithe playe wants to do anything di eent, and it is a Nash equilibium of the game. So, Rock-Pape-Scissos has a Nash equilibium, but only when both playes ae using just the ight mixed stategies. To detemine the payo that two playes get when they ae both using mixed stategies, we use the expected utility as befoe, but must now combine both playes stategies. If Playe 1 has n pue stategies and plays a mixed stategy 1 = (x 1,x 2,...,x n )andplaye2hasmstategies and uses a mixed stategy 2 = (y 1,y 2,...,y m ), then Playe 1 s payo against Playe 2 can be calculated as follows: u( 1, 2) = nx mx u(s i, j )x i y j. i=1 j=1 s i epesents Playe 1 s ith stategy and j epesents Playe 2 s jth stategy. In othe wods, we conside evey possible between the two stategies and multiply the payo of that outcome with the chance that outcome happens. Then, we add up all those possibilities. 4.1 Finding a mixed stategy equilibium in 2 2 games Finding mixed stategy equilibia in games can be vey had, especially if the games have many playes o many stategies. But, fo small games thee ae some ticks we can use to find mixed stategy Nash equilibia. This section uses an example to demonstate how to find a mixed stategy Nash equilibium in games that have 2-playes with 2-stategies each (sometimes called 2 2games ). Notice that when we found the equilibium fo Rock-Pape-Scissos, it was when both playes wee using stategies that made the othe playe not cae what which stategy they played. This tuns out to be a geneal fact about the mixed stategies that make up Nash equilibia. So, if we can figue out which combination of stategies make playes not cae which pue stategy they play, that will tell us whee the equilibium is. This is easiest to see with an example, conside the game below. 9
P laye2 P laye1 c d a 0, 0 1, 3 b 2, 1 0, 0 Example game #2 This game has thee equilibia. Two ae pue stategy equilibia: (a, d) and (b, c). But, it also has a thid whee if both playes mix between thei stategies just ight, neithe will do bette by doing anything di eently. To find this mixed equilibium, we need to find the stategies that make each playe indi eent. Fist conside the game fom the pespective of P 1. If P 2 plays c with pobability y and d with pobability (1 y), we ask: what value of y makes P 1 indi eent between a and b? TheexpectedutilityfoeachofPlaye1 sstategiescanbedetemined as follows: EU 1 (a, y) =P(c)u 1 (a, c)+p(d)u 1 (a, d) =y(0) + (1 EU 1 (b, y) =P(c)u 1 (b, c)+p(d)u 1 (b, d) =y(2) + (1 y)(1) y)(0) Playe 1 is indi eent between his two stategies a and b wheneve the expected utility of his two stategies is the same: EU 1 (b, y) =EU 1 (a, y) o... y(2) + (1 y)(0) = y(0) + (1 y)(1). Solving fo y gives us y =1/3. This means that Playe 1 will be indi eent between he stategies when Playe 2 plays c with pobability 1/3 andd with pobability 2/3. Now we do the same thing, but fom the pespective of Playe 2. Suppose Playe 1 plays a with pobability x and plays b with pobability (1 x). We now ask: what value of x makes Playe 2 indi eent between he two stategies? EU 2 (c, x) =P(a)u 2 (a, c)+p(b)u 2 (b, c) =x(0) + (1 EU 2 (d, x) =P(a)u 2 (a, d)+p(b)u 2 (b, d) =x(3) + (1 x)(1) x)(0) Playe 2 is indi eent when EU 2 (d, x) =EV (c, x) owhenevex(3) + (1 x)(0) = x(0)+(1 x)(1). Solving fo x gives us x =1/4, which means that Playe 2 will be 10
indi eent between he two stategies wheneve Playe 1 plays a with pobability 1/4 andb with pobability 3/4. By combining these two calculations, we can get the mixed stategy equilibium. If Playe 1 uses the mixed stategy 1 =(1/4, 3/4) and Playe 2 uses the mixed stategy 2 =(1/3, 2/3), then both playes will be indi eent between thei two stategies. This means that neithe playe can do bette by switching thei stategy given what the othe is doing. So, the stategy pofile ( 1, 2) is a Nash equilibium. 5 The Equilibium Selection Poblem The game in the pevious section (Example game #2) has thee Nash equilibia: the two pue stategy equilibia (b, c) and(a, d) alongwiththemixedstategy equilibium descibed in the pevious section. Many games, and most of the games that ae eally inteesting, have multiple Nash equilibia. What does game theoy pedict (o ecommend, depending on how we intepet a solution to the game) in these cases? If we want to use game theoy make a specific pediction, o to ecommend a paticula stategy when playing a game, we need to choose which of the solutions is the eal solution. How we choose among the equilibia will, of couse, depend on what we ae tying to use the solutions to epesent. But, no matte what we ae tying to captue, we will need to select which of the equilibia ae the elevant ones. This is called the equilibium selection poblem in game theoy. Conside the Stag Hunt game (below). In this game thee ae thee equilibia: (c,c), (d,d) and a mixed equilibium whee both playes choose c 75% of the time and d the est of the time. So what would game theoy pedict people will do when playing this game when evey action that the playes could take fom pat of some equilibium o othe? Without a way to appoach the equilibium selection poblem, game theoy tells us nothing about what to pedict (o what to do) in this game. P laye2 P laye1 c d c 4, 4 0, 3 d 3, 0 3, 3 The Stag Hunt Thee ae many di eent aspects of the game and its vaious Nash equilibia that we might conside when facing this poblem. Fo example, the (c,c) equilib- 11
ium gives the best possible payo to eveyone. This equilibium is called paeto optimal and is the payo dominant equilibium. But an agument can be made that the othe equilibium, (d,d) is the moe natual solution because it would esult fom both playes avoiding isk. Suppose that both ae eally not sue what the othe will do and think thee is an equal chance that the othe playe will pay any of he stategies. Then, choosing d guaantees the playes a decent playo. The (d,d) equilibium is the isk dominant equilibium. It is these kind of consideations that can help appoach the equilibium selection poblem. Howeve, thee is not a unique solution to this poblem and the best way to appoach it will depend on what we ae tying to epesent using game theoy. As mentioned above, the equilibium selection poblem aises in many games. This is especially tue in games whee seveal moves ae made and the ode of the moves mattes. These games ae di cult to epesent in the way that we have been doing so fa (the nomal fom), but can be epesented in othe ways. 6 Extensive Fom Games Not all games ae best epesented in nomal fom. In some games, one playe moves fist and the othe playe can espond. In these games, the ode of actions makes a di eence and this is not easily epesented in the nomal fom matices. To captue games whee the sequence of actions is impotant, we use what is called an extenstive fom epesentation. Conside the chain-stoe game. In this game, one company (P1) must choose between enteing a new maket (e) o not (n). If they choose not to, then eveything emains as it is. If they choose to ente the maket, anothe company (P2) must choose whethe o not to challenge (c o d). If P2 decides not to challenge, the playes split the maket. If P2 decides to challenge, they must pay a heavy cost since both playes will be foced to sell vey cheaply o lose to the othe playe. This game is best epesented in the extensive fom (see Figue 1). Which captues the sequence of move in a tee-fom. The game can also be expessed in nomal fom (see below), but this fom does not captue the fact that the moves ae made in a cetain sequence. P 1 P 2 c d e 1, 1 2, 2 n 0, 4 0, 4 The Chain-Stoe Game (nomal fom) 12
0, 4 n Pb1 @ @ e @ @ P 2 @ c @ d @@ 1, 1 2, 2 Figue 1: The chain-stoe game in extensive fom Examining the nomal fom epesentation of the chain-stoe game, we can see that thee ae two pue-stategy Nash equilibia: (e, d) and(n, c). The fist of these is whee P1 entes the maket and P2 gives up half of it. The second is whee P1 does not ente the maket and P2 would challenge if P1 wee to ente the maket. Notice that thee is something a little stange about the (n, c) equilibium if P1 eve actually chose e, it wouldn t make sense fo P2 to chose c, as they would be choosing -1 ove 2. It only makes sense fo P2 to choose c if P2 neve has to make a choice! This means that the (n, c) equilibiumismaintained by a theat that wouldn t be ational to cay out if it came down to it. This equilibium is not subgame pefect (see the next section). We can also epesent uncetainty within extensive fom games. And, we can use this uncetainty to epesent simultaneous moves using the extensive fom. Conside the Diving Game (fom above) whee =Rightandl =Left.Wecan suppose that Playe 1 does move fist, but that Playe 2 simply does not know what Playe 1 chose. The dotted line between the two choice points indicates that P2 does not know which choice point she is located. l 1, 1 P1 b HHH l H pppppppppppppppppppp P2 HH @ l @ @ @ @ @ 0, 0 0, 0 1, 1 Figue 2: The Diving Game in extensive fom One famous game that is best expessed in extensive fom is the Ultimatum Game. Suppose that thee ae two playe that must divide 10 units of some good: the Popose and the Responde. The Popose poposes a split to the Responde, who can accept o eject the split. If the Responde accepts, each gets thei shae of the poposed split. If the Responde ejects, each gets nothing. To simplify things, we can suppose that the Popose only has two possible poposals 13
to make: an unfai 9-1 split o a fai 5-5 split. This simplified game is epesented in Figue 3. Popose b @ @ fai @ unfai @ Responde Responde @ @ accept @ eject accept @ eject @ @@ @ 5, 5 0, 0 9, 1 0, 0 Figue 3: A Simplified Ultimatum Game A stategy in an extensive fom game has to tell the playe what to do at evey single choice point that they might make. So, the Popose s stategy set is {fai, unfai}. The Responde s has fou possible stategies, which specify what she would say to each possible o e: {accept both, accept fai/eject unfai, eject fai/accept unfai, eject both} o {aa, a, a, } fo shot. Thee ae thee pue-stategy Nash equlibia of this game: (unfai, accept both), (unfai, eject fai/accept unfai), and (fai, accept fai/eject unfai). This is easiest to see if we epesent the game in nomal fom (below). Responde Popose aa a a fai 5,5 5,5 0,0 0,0 unfai 9,1 0,0 9,1 0,0 Simplified Ultimatum Game (nomal fom) 7 Subgame Pefection Two of the equilibia in the simplified Ultimatum Game have a athe stange featue if they ae to be consideed ational solutions. Both the (unfai, eject fai/accept unfai), and (fai, accept fai/eject unfai) equilibia involve ejections whee the Responde e ectively chooses nothing ove something. The only eason that these ae equilibia is just that the Responde neve actually has to make that choice given what the Popose is o eing. But, if the Popose wee to eve o e anything di eent, it wouldn t be in the Responde s inteest to eject the o e (something is always bette than nothing). 14
So, these equilibia ae maintained by the theat of a possible futue action that is iational. Some games, such as the simplified Ultimatum Game, contain what ae called subgames. A subgame is a game-within-a-game. The simplified Ultimatum Game (in Figue 3) has thee subgames: (i) the whole game, (ii) the fok in the tee on the left side beginning at the Responde s choice, and (iii) the fok in the tee on the ight side beginning at the Responde s choice. Using this idea, we can define a moe ational vesion of the Nash equilibium: the subgame pefect equilibium. Subgame Pefect Equilibium: A Nash equilibium of a game that also foms a Nash equilibium of all the subgames. In the case of the simplified Ultimatum Game, thee is only one subgame pefect equilibium: (unfai, accept both). Thinking of the two small subgames, it is iational fo the Responde to choose eject in eithe case. And, given that it is always ational fo the Responde to accept, it is ational fo the Popose to o e the unfai split. In the othe two Nash equilibia of the game, the Responde has a stategy that does not constitute a Nash equilibium of one o othe of the subgames. The idea of a subgame pefect equilibium is one way that we might go about solving the equilibium selection poblem in extensive fom games. This might be of special inteest if we want ou solutions to captue the ational outcomes of agame. 7.1 Backwad Induction One common method fo finding a subgame pefect equilibium is called backwad induction. This method begins by looking at the vey end of the game and asking: if this decision point wee eached, what would a ational playe do? Then imagining that the game ends with that ational decision, you move on to the pevious decision in the game and ask again: given that the last choice will be ational, what is the ational choice to make at the second-to-last choice points? And so on, to the beginning of the game. Pb1 P 2 R P 1 P 2 R P 1 P 2 R 6, 6 d D d D d D 2, 0 1, 3 4, 2 3, 5 6, 4 5, 7 Figue 4: The Centipede Game 15
Conside the Centipede Game (Figue 4). In this game, playes take tuns choosing whethe o not to take the majoity of a pool of esouces and end the game o to pass the choice to the othe playe, theeby inceasing the pool of esouces. The game continues fo a fixed numbe of ounds, if no one takes the majoity, the game ends with an equal split. Accoding to the method of backwad induction, lets begin at the end of the game. At the end of the game, Playe 2 should choose to take the majoity and end the game (play D). Given that playe two should do this, Playe 1 should take the majoity one ound soone and end the game. Given that Playe 1 should do this at the second-to-last point, Playe 2 should take the majoity at the the thid-to-last-point, and so on. The subgame pefect equilibium is fo both playes to take the majoity at evey possible choice point: (ddd, DDD). 16