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Game Theory, Overview

Ram C. RaoUniversity of Texas, Dallas, Richardson, Texas, USA

Glossary

asymmetric information The type of data acquired whena move by Nature occurs before any player acts, and some,but not all, players are ignorant of what Nature chose.

best response A player’s optimal choice of action in responseto a particular choice of others.

common knowledge Something that all players know, withthe shared understanding among all players that they allknow the same thing.

contractible Some element that can be written into a contractso that the disposition of the contract is based on the exactoutcome of the element, and a third party, other than theplayers, can help in verifying the outcome.

cooperative game An activity in which players can makebinding agreements as to their actions.

equilibrium and Nash equilibrium A set of actions chosenby all players such that, given the choices of others, noplayer wants to change his choice unilaterally.

folk theorem A result, especially in repeated games, thatmany hold to be true, but which is proved formally muchlater.

incomplete information The type of data acquired whena move by Nature occurs before any player acts, and someplayers do not know what Nature chose.

informed player A game participant who has observedNature’s move in a game of asymmetric information.

mixed strategy A player’s choice of action that attachesa probability to more than one available action.

Nature An entity that makes a probabilistic move with respectto one of the elements of the game.

noncooperative game A type of game in which playerscannot make binding agreements as to their actions.

payoff Profits or utility to players after all players have chosentheir action.

perfect information The type of data acquired when playersknow all that has occurred before their move.

players Participants in a game.pure strategy A player’s choice of a single action with

probability 1, from among all available actions.

strategy Specifies action for a player at each information setin which he must move.

subgame perfect Nash equilibrium A Nash equilibrium toa game that is also a Nash equilibrium in every subgame.

Game theory may be used as a modeling device. Therudiments of noncooperative game theory includeanalyses of games of incomplete information, emphasiz-ing the concepts of Nash equilibrium, subgame perfec-tion, and perfect Bayesian equilibrium. Both illustrativeexamples and applications in economics and managementare used here to provide the reader with a grasp ofthe scope and usefulness of the game theoretic model.Also presented are the challenges facing modelers whowant to use game theory, and how some of those are beingmet by continuing research.

Game Theory as a Model

Why Game Theory?

Game theory can be viewed in terms of its mathematics oras a tool to model the interaction between decision mak-ers. The word ‘‘game’’ is an apt one to describe this be-cause, just as in common parlor games such as Chess orHex, much of game theory is concerned with how indi-vidual entities (persons, or organizations) choose actions,taking into account how other participants do the same.These entities are called players, even though the deci-sions that they make are in the context of situations withreal-world consequences, quite different from the enter-tainment that parlor games yield. Players are assumedto choose actions to maximize their expected utility,following the accepted model of single-person decisionmaking. In this sense, game theory may be viewed as

Encyclopedia of Social Measurement, Volume 2 �2005, Elsevier Inc. All Rights Reserved. 85

a generalization of single-person decision theory tomultiperson decision making. But there are manydifferences between the two. The utility resulting froma player’s action cannot be determined without also takinginto account the actions chosen by other players. Thus,game theory cannot prescribe an optimal action for anindividual player without also offering a way for eachplayer to anticipate what other players would choose.In other words, game theory is concerned with specifyingactions for all players, ensuring that for each player, his/her chosen actions are optimal, given the actions of otherplayers, implying that optimality is relative. As a result, it isgenerally difficult to define the best outcome from theview of all players. The value of game theory, then, lies inits ability to model the interaction between players.Such a model can help to explain observations involvingmultiperson decision-making situations, and can alsorule out certain outcomes that might not otherwisebe contemplated. There is another potential use ofgame theory. If one of the players can ‘‘go’’ (act) firstand choose an action that he can commit to, then it ispossible to come up with choices that will ensure out-comes favorable to the first player. For example, thetax code can be viewed as a choice of lawmakers.Based on that, citizens and corporations choose actions,given the tax code. Thus, the lawmakers and citizens canbe viewed as playing a game. An analysis of such a gamecan help determine the best code from the lawmakers’viewpoint. And, what is more important, it would help todisabuse wishful lawmakers of predictions of outcomesthat are deemed unreasonable in light of the analysis.

Questions

What should a person know about game theory to beable to apply it to problems in social science? This ques-tion can be answered using applications and illustrativeexamples, mainly in economics and management, that donot require advanced training. The deeper problems ineconomics and management that game theory has beenused to analyze can be accessed in a large body of liter-ature. The hard problems in game theory are not dis-cussed here, yet the reader can get a feel for how thegame theoretic model has been successfully used basedon minimal domain knowledge beyond everyday experi-ence. Issues that some would regard as drawbacks inusing game theory are presented here, although others,including the author, view them as challenges.

Definitions

A game, denoted by G, can be defined as consisting ofthree elements: players, indexed by i (i¼ 1, 2, . . . , N); anaction or strategy ai, possibly a vector, chosen by player i,from a set Ai¼ {ai}; and a payoff to player i, pi(ai, a�i),

where a�i denotes the actions of all players other than i.The payoff pi is to be thought of as the utility to player iwhen he chooses ai and the other players choose a�i.A game defined in this way is represented in strategicform. An alternative representation of a game, called ex-tensive form, is illustrated by considering a game such aschess. In chess, each player has a turn to make a move bychoosing an action ai, followed by other players, each inturn. The sequence of decisions forms a decision tree, ormore correctly a game tree. Each turn to move by anyplayer can be thought of as a stage in a multistage game.A multistage game is to game theory what dynamic pro-gramming is to single-person decision theory. Ina multistage game, when it is a player’s turn to move,he/she may or may not know what some or all of others’choices are. Depending on what the player knows, he/sheis at a particular information set at each turn to move.Viewed this way, player i’s strategy, si, is defined asa specification of an action, ai, at each information set,satisfying si[ Si. Payoffs can be denoted in this case by pi(si, s�i), si[ Si. In chess and other games, there could bea chance element, similar to the toss of a coin, that de-termines who moves first. Any such chance event is as-cribed to what is denoted ‘‘Nature.’’ Finally, assume thateach player knows the payoffs and the structure of thegame, and further that each player knows that the otherplayers know the same information, and also knows thatother players know that the other players know, and soon, ad infinitum. This type of knowledge of the game byall of the players is known as common knowledge.

Classification of Games

There are many ways to classify games. First, a game canbe a cooperative or noncooperative game. In a cooperativegame, the players can make binding commitments withrespect to their strategies. Thus, members of the Orga-nization of Petroleum Exporting Countries (OPEC), forexample, play a cooperative game when they agree to whatlevel of oil production each is to maintain. In contrast,supermarkets, say, choose pricing strategies noncooper-atively. Here the concern is only with noncooperativegames.

A very important basis for classifying games is theinformation that players have at the time of makinga move. A game is said to be one of perfect informationif, at his/her turn to move, each player knows all that hasoccurred up to that point in the game. Chess is anexample of this type of game. In contrast, considertwo firms submitting sealed bids on a contract. Thiscan be conceptualized as similar to a game of chess,with one firm moving first and submitting a bid, followedby the other. Naturally, the first player does not knowwhat the other will do. But what is more important isthat the second player does not know what the first bid is.

86 Game Theory, Overview

The second player’s information set consists of allpossible bids that the first player could have made.Stated differently, the second player’s information setcontains many elements, each element corresponding toa possible bid by the first player. A game is said to be oneof perfect information if, at each player’s turn to move,his/her information set is a singleton. With this defini-tion, it is easy to see that bidding for the contract is nota game of perfect information. When players movesimultaneously, as in this case, the game is always oneof imperfect information. An even more importantdistinction arises if Nature moves first, and the moveis not revealed to at least one player. For example, sup-pose a human plays chess against a computer but doesnot know whether the computer is playing at the begin-ner or advanced level. When the human sees a particularmove by the computer, he/she must recognize that themove could have resulted from either the beginner orthe advanced level of playing. The player’s informationset is therefore not a singleton, even though he/sheobserved the move. In fact, it can be said that thehuman does not know which game he/she is playing;chess against beginner level or chess against advancedlevel. This is called a game of incomplete information.More generally, in games of incomplete information,one or more players do not know the game structure.Naturally, all games of incomplete information are alsogames of imperfect information. A special case of a gameof incomplete information is one of asymmetric infor-mation. In a game of asymmetric information, the moveby Nature is observed by some players but not by others.Consider the interaction between a car salesperson anda potential buyer. The buyer knows his/her budgetbut the salesperson does not. Thus, in the game ofhaggling, the salesperson’s information is differentfrom that of the buyer, and so there is a game of asym-metric information. As far as the salesperson isconcerned, the buyer is a random draw, attributableto a move by Nature. And so, it is a game of incompleteinformation.

Equilibrium

Nash Equilibrium

In single-person decision theory, the main task is to char-acterize the optimal solution. In game theory, the task is tocharacterize the equilibrium. The equilibrium is under-standable if the concept of best response is first under-stood. Consider player i’s choice in response to actionsby the other players. Denote bi(s�i) to be i’s best responseto s�i. It is then defined as

biðs� iÞ ¼ arg maxai[Ai

pi si, s� ið Þ: ð1Þ

An equilibrium s� to a game is defined as the strategiesof all players such that its components (si�, s� i

� ) satisfy thefollowing relationship:

si� ¼ biðs�1

� Þ 8i ð2Þ

In other words, at equilibrium, each player’s choice isa best response to the choices of the remaining players.In this sense, each player’s choice is the solution to anappropriately defined optimization problem. Becauseeach player is at his best response, each player is contentwith his/her choice. For this reason, this is called anequilibrium. Another equivalent way to define theequilibrium choices s� is to impose the followinginequalities at the equilibrium:

piðsi�, s�i� Þ � piðsi, s�i

� Þ si [ Si 8i ð3Þ

An equilibrium s� is usually called a Nash equilibrium,named after John Nash. The essential property of Nashequilibrium is that once players arrive at such anequilibrium, no player can profit by deviating unilat-erally from it.

Pure and Mixed Strategies

Thus far, strategies have been viewed as actions, or rulesfor action, at each information set of a player. The defi-nition of strategies can be broadened by specifyinga strategy as a probability function over the actionsai[Ai corresponding to each information set of theplayer. A nondegenerate probability distribution wouldimply that the player’s strategy tells him/her to mix amonghis/her actions, rather than have a unique action fora choice. For this reason, strategies defined in this wayare called mixed strategies. A simple example will helpillustrate the idea of mixed strategies. Consider a baseballpitcher pitching to a new batter. Suppose the pitcher isrestricted to pitch a strike (meaning that the ball goes overthe plate at the right height for the batter to be in a positionto hit) or a ball (meaning that the ball goes far away fromthe batter, making it difficult to hit). So the pitcher hastwo possible actions from which to choose. Now, supposethe batter also has two options: to take a swing or to‘‘see’’ the ball, meaning not to take a swing. If the batterswings at a strike or sees a ball, the batter wins, whereasthe pitcher wins if he/she throws a strike and the battersees, or he/she throws a ball at which the batter swings. Ifthe pitcher decides to throw a strike for certain, the batterwould swing and win. Likewise, if the pitcher decidesto throw a ball for certain, the batter would see andwin. The pitcher may then consider mixing pitches sothat there is uncertainty as to whether he/she throwsa strike or a ball. In other words, the pitcher may chooseto employ a mixed strategy. In contrast, if the pitcherchooses to assign a probability of 1 to either of his/heractions, he/she would be employing a pure strategy. Thus,

Game Theory, Overview 87

a pure strategy is a special case of mixed strategies inwhich all actions but one have a zero probability ofbeing chosen.

One advantage with broadening the definition of strat-egies in this way is that it assures us of the existence ofa Nash equilibrium, possibly in mixed strategies, for anygame G with a finite number of players, each of whoseaction sets contain a finite number of elements. John Nashfirst established this existence result in 1950.

Equilibrium in Pure andMixed Strategies

Examination of a few game structures will help illustrateNash equilibrium in pure and mixed strategies. Three ofthese structures are celebrated, both for their ability toilluminate key concepts and because they representgeneric models of competition. All three games havetwo players, and each player has two possible coursesof action.

Product Introduction (Prisoners’ Dilemma)This game contemplates two firms who may each intro-duce (I) or shelve (S) a new product (P). Table I shows thepayoffs to each firm under the four possible combinationsof actions. In the table, the rows represent the possibleactions of firm 1, and the columns represent the possibleactions of firm 2. The entries in each cell are the payoffs tothe two firms. Thus, if both chooseS, both are at status quowith a utility of 0; if both choose I, they incur costs, but nogains, with utility of �2. Consider first the best responsesof firm 1. It is obvious that b1(S)¼ I and b1(I)¼ I. In otherwords, regardless of what firm 2 chooses, the best re-sponse of firm 1 is to choose I. For firm 1, the strategyS is strictly dominated because it is inferior in both sce-narios (with firm 2 choosing S or I). The dominated strat-egy S for firm 1 can therefore be eliminated or deleted. Bysymmetry, S for firm 2 can also be deleted. This leavesa unique outcome arising from the dominant strategy of Ifor each firm. The Nash equilibrium for this game then is(I, I), as indicated by (�2, �2) in Table I. It can be seenthat the Nash equilibrium is in pure strategies. Note alsothat the Nash equilibrium is unique. Finally, it yieldslower utility to both firms than what they could achieveby cooperating and choosing S. Of course, then they

would not be at a best response. Though the best responseensures optimality in an appropriately defined maximiza-tion problem, the maximization problem may be undesir-able if it is compared to what is possible under cooperativebehavior, a feature that is a property of many noncoop-erative games. This game is often referred to as theprisoners’ dilemma, replacing firms by prisoners, actionsS and I by ‘‘deny’’ and ‘‘confess’’ (to a crime), and leavingthe ordering of the utilities the same. In that game, ‘‘con-fess’’ is a dominant strategy for both, but if they couldcooperate to choose ‘‘deny,’’ they would be better off. Thiswill be discussed further later.

Battle of the SexesThe next illustrative game is called the battle of the sexes.In this game, the two players are a man and a woman.Each can choose to go to the ballet or to the baseball game.The payoffs to the two players are shown in Table II. In thetable, when both the man and the woman choose base-ball, the man gets a utility of 2 and the woman gets a utilityof 1, and so on. It is easy to see that the man’s best re-sponses are given by bM(baseball)¼ baseball and bM(bal-let)¼ ballet. Likewise, the woman’s best responses arebW(baseball) ¼ baseball and bW(ballet) ¼ ballet. Thus,there are two Nash equilibria in pure strategies in thiscase. The first one has both players choosing baseball, andthe second one has both choosing ballet. An importantpoint to note here is that the Nash equilibrium is notnecessarily unique. Moreover, whereas the man wouldprefer the first equilibrium—he obtains a utility of 2 asopposed to 1 in the second—the woman would prefer thesecond equilibrium. Thus, it is not possible, in general, torank order Nash equilibria in terms of desirability. Finally,it is not clear how the two players can agree on one of thetwo equilibria if they cannot communicate in some way.Perhaps even talk that is nonbinding can help the two toget to one of the pure strategy Nash equilibria. Such talkthat is nonbinding is termed ‘‘cheap talk.’’

Monitoring a Franchisee (InspectionProblem)The final example will help illustrate the Nash equilibriumin mixed strategies. In this example, a franchisor maychoose to monitor the performance of a franchisee ormay choose ‘‘don’t monitor.’’ The franchisee can chooseshirk or ‘‘don’t shirk.’’ The payoffs are shown in Table III.

Table I Product Introductiona

Firm 2 (P2)

Firm 1 (P1) S I

S (0, �0) (�5, 10)

I (10, �5) (�2, �2)

aProduct P is either introduced (I) or shelved (S).

Table II Battle of the Sexes

WomanMan Baseball Ballet

Baseball 2, 1 0, 0

Ballet 0, 0 1, 2


This game is also referred to as the ‘‘inspection problem,’’for obvious reasons. Once again, the best responses of thefranchisor can be seen to be bFR(shirk) ¼monitor andbFR(don’t shirk)¼ don’t monitor. And the best responsesof the franchisee are bFE(monitor)¼ don’t shirk and bFE

(don’t monitor) ¼ shirk. It is obvious that there is no purestrategy Nash equilibrium by an examination of the bestresponses. For example, taking the outcome (monitor,shirk), though the franchisor is at a best response, thefranchisee is not. Similarly, all four possible outcomescan be ruled out. Although there is no Nash equilibriumin pure strategies in this case, there exists a Nash equi-librium in mixed strategies. Let p denote the probabilitythat the franchisor chooses monitor and let q denote theprobability that the franchisee chooses shirk. Supposeq¼ 0.5. Then, the franchisor’s expected profits are0.5� (�1)þ 0.5� (�3)¼�2, if he/she chooses monitor,and 0.5� (�4)þ 0.5� (0)¼�2, if he/she chooses don’tmonitor. In other words, the franchisor is indifferentbetween his/her two actions if the franchisee followsthe mixed strategy of q¼ 0.5. Because the franchisor isindifferent between the two actions, he/she can random-ize across them. Suppose the franchisor chooses themixed strategy p¼ 0.5. Then, it can be verified that thefranchisee is indifferent between his/her two actions.The franchisee could, therefore, choose to randomizeacross his/her actions, and indeed choose q¼ 0.5. Thus,it is obvious that the pair (p¼ 0.5, q¼ 0.5) is a best

response for each player and so constitutes a Nashequilibrium in mixed strategies. The key point to notehere is that in mixed-strategy equilibrium, a player is in-different across actions chosen with nonzero probability.Indeed, each of these actions is a best response to theother players’ equilibrium mixed strategy. Moreover, if anaction is not a best response at equilibrium, it is chosenwith zero probability. Finally, the mixing probabilities ofthe other players make each player indifferent across ac-tions chosen with nonzero probability. In fact, the battle-of-the-sexes game also a mixed-strategy equilibrium inaddition to the two pure-strategy equilibria.

Informational Issues

Perfect Equilibrium

One of the central ideas in dynamic programming isBellman’s principle of optimality. This has force in mul-tistage games in the following sense: a player’s strategyshould be such that it is a best response at each informa-tion set. To see how this might affect Nash equilibrium,consider the following variation of the battle of the sexes.

Subgame PerfectionSuppose in the game of battle of the sexes, the man movesfirst in stage 1 and makes a choice, followed by the woman,in stage 2, and the man gets to have the final say in stage 3.This multistage game can be represented in extensiveform by the game tree in Fig. 1, similar to a decisiontree in single-person decision situations. At the end ofeach terminal branch in the game tree are the payoffsto both of the players. Read Fig. 1 as follows: If instage 1, at node 1, the man chooses baseball (BS), thisleads to node 2 in stage 2 of the tree. If, on the other

Table III Monitoring a Franchisee

FranchiseeFranchisor Shirk Don’t Shirk

Monitor �1, �1 �3, 0

Don’t monitor �4, 1� 00, 0

1

Man

BS BL

Woman 2 3 Woman

BS

BS

BL BS BL

4 Man 5 Man Man 6 Man 7

BL BS BL BS BSBL BL

Payoffs Man Woman

2 0 0 1 2 0 0 11 0 0 2 1 0 0 2

Figure 1 A three-stage battle-of-the-sexes game in extensive form. BS, baseball; BL, ballet.


hand, the man chooses ballet (BL), this leads to node 3 instage 2. At node 2, if the woman chooses ballet, this leadsto node 5 in stage 3, and the man would choose again. Atnode 5, the best that the man could do is to choose balletand get a payoff of 1, as opposed to a payoff of 0 bychoosing baseball. Note that node 5 would give a payoffof 2 to the woman, and, using the same logic, node 4 wouldgive her 1. Knowing this, at node 2, she would chooseballet. And similarly, at node 3, she would choose ballet.Given this, at node 1, the man could choose either ballet orbaseball. The Nash equilibrium outcome to the three-stage battle of the sexes is unique: both the man andthe woman go to the ballet. The equilibrium is specifiedas follows:

Man: Stage 1—at node 1, choose either ballet or base-ball; stage 3—at nodes 4 and 6, choose baseball and atnodes 5 and 7 choose ballet.

Woman: Stage 2—at nodes 2 and 3, choose ballet.

One of the interesting things about specifying the equi-librium in this way is that even though, in equilibrium,nodes 4 and 6 would never be reached, what the manshould do if the game ever got there is specified; i.e.,the man’s strategies are specified on off-equilibriumpaths. Also, note that in solving for the equilibrium tothe game starting at node 1, the games are first solvedfor starting at nodes 4�7 in stage 3, then these strategiesare used from stage 3 to solve for games starting fromnodes 2 and 3 in stage 2, resorting to backward inductionto solve for the game. In doing so, the equilibrium is solvedfor all subgames, starting from every node. Thus, theequilibrium specified is a Nash equilibrium, not only tothe game starting from node 1, but to every subgame. Forthis reason, the equilibrium is known as a subgame perfectNash equilibrium. It is important to note that if sub-gameperfection was not required, other Nash equilibria would

be admitted, including, for example, the man choosingbaseball at nodes 1 and 4�7, and the woman choosingbaseball at nodes 2 and 3. Finally, note that if the three-stage game had only two stages, containing nodes 1�3,with the man choosing in stage 1, at node 1, and thewoman choosing in stage 2, at nodes 2 and 3, thesubgame perfect equilibrium would be for both to choosebaseball. In this way, it is seen that the concept of subgameperfect equilibrium has real force. It has been used tostudy many problems in economics, especially relatedto entry of firms, including sequential and preemptiveentry. Essentially, the idea is that once a firm has entered,an incumbent would take that as given, and the subse-quent behavior would be an equilibrium to the new sub-game postentry, and it is this that must govern a firm’sentry decision.

Incomplete and AsymmetricInformation

Recall that if there is a move by Nature—a chance move—at the beginning of a multistage game, then the game isone of incomplete information. Consider the followingexample. A seller of an appliance, say, a refrigerator,knows that there is a probability p that the refrigeratorhas a defect that cannot be observed. Suppose the buyeralso knows this. In other words, the defect rate is commonknowledge. The buyer would pay $1000 for a ‘‘good’’ re-frigerator and $900 for a ‘‘bad’’ one. Should the seller seta price, P, of $1000 or $900? (for the sake of exposition,ignore other possible prices). And, what would be theequilibrium action for the buyer: buy, B, or not buy,N? This is a game of incomplete information, whichcan be depicted as in Fig. 2. Note that the seller’s infor-mation set consists of the two nodes {1, 2} because he/she does not know whether Nature chose a good

1Nature

Good Bad1– p p

Seller2 3

$900 $9005 Buyer 7

P = $1000 P = $1000Buy Don’tbuy

Buy Don’tbuy

Don’tbuy

Buy

Don’tbuy

Buy

4 6Buyer

Figure 2 Game of incomplete, symmetric information. The dashed lines indicatethe same information set.


refrigerator or a bad one. If the seller chooses a price of$1000, the buyer is at the information set consisting of thetwo nodes {4, 6}, and if the seller chooses a price of $900,the buyer is at {5, 7}. This game might be solved by in-voking, for example, the expected value criterion. What isimportant to note here, though, is that in evaluating thebest response based on expected value at an informationset with multiple nodes, a decision maker must know thepayoffs for any combination of node and action and mustknow the probability of being at a certain node condi-tioned on being at the information set. Figure 2 repres-ents a game in which information is incomplete andsymmetric, because neither seller nor buyer knows whatNature chose.

Now consider games of asymmetric information,which provide useful models for many interesting situa-tions. Returning to the refrigerator example, suppose itcan be one of two types with a defect rate ofpi, i¼ 1, 2, andp15p2. So, product 1 is good and product 2 is bad. More-over, assume that buyers value a working refrigerator at$1000 and a defective one at zero. Further, buyers are riskneutral, so they would pay $1000(1� p) for a refrigeratorwith a known defect rate p. The cost to the seller of a type irefrigerator is assumed to be Ci. Thus, the profits fromselling type i refrigerator, if buyers were informed ofthe type, is 1000(1 � pi)�Ci. Finally, assume that theprobability that the seller has a good refrigerator is y,and bad is 1� y. If consumers had no information onthe type that the seller has, they would be willing topay 1000[(1� p1)yþ (1� p2)(1� y)]. Let P denotethe price that the seller sets. Then, the buyer�seller in-teraction is depicted as a game of asymmetric informationin Fig. 3. Here, as in Fig. 2, Nature picks the type ofrefrigerator at node 1, the seller picks a price at nodes2 and 3, and the buyer makes a decision to buy or not at aninformation set consisting of nodes 4 and 5. The key pointto note here is that the seller knows the node at which he/she is located but the buyer does not, resulting in asym-metric information. The more interesting question is what

the seller can do about it. There are two possibilities. Oneis for the seller to condition his/her equilibrium strategyon the node where he/she is located. This, in turn, wouldreveal information to the buyer, and so the buyer shouldrevise his/her probability of a good refrigerator y (being at4 or 5) suitably, using Bayes’ rule. The buyer’s equilibriumstrategy at the information set containing nodes 4 and 5should then be arrived at using these revised probabilities.The second possibility is that the seller’s equilibrium strat-egy is independent of the information he/she has, and so ofthe node where he/she is located. In this case, the buyerneeds to make sense of an off-equilibrium outcome instage 1. In particular, how should the buyer revise his/her probability of a good refrigerator? As will be seen, thishas force in perfect Bayesian Nash equilibrium (PBNE),which extends the concept of subgame perfect Nash equi-librium to the case of asymmetric information.

Perfect Bayesian Nash Equilibrium

Perfect Bayesian Nash equilibrium is now illustrated toshow how it applies in a simple signaling model. Revisitthe refrigerator example with specific numerical values:p1 ¼ 0.1, p2 ¼ 0.2, y¼ 0.5, C1 ¼ 845, and C2 ¼ 750. Then,consider the following candidate PBNE:

Seller: P¼ 1000[(1� p1)yþ (1� p2)(1� y)]¼ 850.Buyer: Buy.Beliefs (off-equilibrium): Pr(Good |P4 6¼ 850)¼ y.

Given the strategies of the other, both the buyer and theseller are at their best response. However, for a subgameperfect equilibrium to this two-stage game, it should bespecified what the buyer would do in stage 2 if theseller were to deviate from the equilibrium price of$850 in stage 1. In a game of asymmetric information,it is specified how the buyer would revise his/her priorprobability of a good refrigerator, y, in this case asPr(Good jP 6¼ 850)¼ y. These are known as beliefs. Thesubject of what beliefs are appropriate has preoccupiedgame theorists considerably. In this case, what are knownas passive conjectures have been proposed, so that onseeing an out-of-equilibrium outcome, the posterior isthe same as the prior. Given this, it is easy to verifythat this is indeed a PBNE. In this equilibrium, bothtypes of sellers take the same equilibrium action, andso cannot be separated based on their actions. For thisreason, the equilibrium is called a pooling equilibrium.Note that in this equilibrium, a bad product makes a largeprofit, (850� 750¼ $100), and a good one makes a smallprofit (850� 845¼ $5).

It might be conjectured that there are other equilibriain this case, because consumers would be willing to pay$900 for a good refrigerator if there were no asymmetricinformation. In particular, is there an equilibrium inwhich the seller charges a high price in node 2 for

1Nature

Good Bad1– p p

Seller 2 3 Seller

P P

Buyer4 5

Don’t buy

Don’tbuy

Buy Buy

Figure 3 Game of incomplete, asymmetric information. Thedashed lines indicate the same information set.


a good refrigerator and a low price in node 3 for a badrefrigerator? An equilibrium of this type that identifies thetype of seller by the seller’s actions is known as a separatingequilibrium. It turns out that a separating equilibriumdoes not exist in this case. To see why, suppose it did.Then, on seeing a high price, the buyer would have toconclude that the refrigerator is good, and so be willing topay a higher price. But then, why would a seller with a badrefrigerator not mimic the good type? Indeed, in this case,the seller would. Thus, for a separating equilibrium toexist, it must be that no type wants to mimic the other.

Next consider how the seller can profitably enforcea separating equilibrium by designing a contract beforesetting the price. This will make it not profitable for thebad type to mimic the good type. Specifically, suppose theseller can offer a money-back guarantee that costs $25to administer. Consider the following PBNE:

Seller: At node 2, offer a money-back guarantee,but not at node 3.Seller: P¼ 1000 at node 2, $800 at node 3.Buyer: Buy.

Now, given the money-back guarantee, the buyer wouldpay $1000 regardless of the defect rate. The buyer wouldcertainly pay $800 even for the bad product. Sothe buyer’s strategy is a best response. For the seller, atnodes 2 and 3, profits are, respectively, 1000(0.9) �25(0.1)� 845¼ $52.5, or 800� 750¼ $50. It is obviousthat the good seller would not want to mimic the bad, buthow about the other way around? If, at node 3, the sellerwere to offer the same terms as at node 2, his/her profitswould be 1000(0.8)� 25(0.2)� 750¼ $45. It is clear thatthe seller would not want to mimic the action at 2. Thus,a way has been devised for the informed player, the seller,to communicate his/her private information to the buyer,the uninformed player. This was done by a signal of qualityin this case. Interestingly, the signal alone did not affectquality. The signaling model as a way to offset asymmetricinformation is used in many contexts, including job mar-ket signaling, if worker’s abilities are not known toemployers; advertising, if product quality is known toseller but not buyer; and dividend payments, to conveyinformation on a company’s future prospects.

The Revelation Principle

A large class of problems in which there are two players,one informed and the other uninformed, has come to beknown as principal�agent problems. A principal is onewho designs a contract that an agent can accept or reject.From a game theoretic perspective, the key question iswhether the informed or the uninformed player movesfirst. In the signaling model, the informed player movesfirst. In the particular example of the preceding section,the principal was informed. Signaling can also occur in

situations in which the agent is informed. Employers(principals) are uninformed of workers’ (agents’) abilities,but workers are informed. In this case, the employer de-signs the contract conditioned on a signal from theworker. Workers accept or reject the contract dependingon their abilities.

Consider now situations in which the uninformedplayer must move first. In particular, assume that theprincipal is uninformed and must design a contract.There are many examples of this kind of situation. Forexample, insurance contracts are designed without know-ing the driving characteristics of the insured. For illustra-tive purposes, consider a sales manager as the principalwho must design a compensation plan (contract) for his/her salespersons, who are the agents. Assume that theagent in this model is informed. To what informationmight a salesperson, but not the manager, have access?It could be the salesperson’s ability. In this case, the sales-person would be informed before accepting the contract.Another kind of information would be whether a customerthe salesperson approached turned out to be easy or hardto sell to, because it would be difficult for the manager toobserve this sales response. In this case, the salespersonbecomes informed after accepting the contract but beforeputting in the effort to sell to the customer. Thus, the effortlevel of the agent depends on information that he/she, butnot the principal, has. Finally, the manager would be un-informed about the effort put in by a salesperson. Thiswould occur if sales have a random component to them,because knowing sales is not sufficient to infer the effort.More relevant, a manager, even if he/she could observethe salesperson, cannot write a contract that is condi-tioned on a salesperson’s ability, the random customerresponse, or effort. Thus the type (of salesperson), thestate of the world chosen by Nature, and effort are allnot contractible, meaning that there would be no wayfor a third party, such as a court, to verify the claims ofability, customer response, and salesperson effort. And yetall of these factors would affect the sales generated by thesalesperson. What the manager can do is to write an ‘‘op-timal’’ contract conditional on sales. What are some prop-erties that such optimal contracts must satisfy?

A contract based on sales (verifiable outcome) mustsatisfy three conditions to be practical. First, it must beacceptable to the salesperson. Usually, it must afford theagent expected utility that is not less than what the agentwould obtain if he/she were to reject the contract. This isknown as the individual rationality (IR) constraint. Sec-ond, it must take into account that the salesperson wouldtake the contract as given and then choose effort to max-imize his/her expected utility. In other words, the incen-tive structure in the contract would determine the effortlevel. This is known as the incentive compatibility (IC)constraint. Third, a contract that is optimal for themanager would maximize his/her expected utility subject


to the IR and IC constraints. This problem can be formu-lated as a constrained maximization problem.

The problem can also be viewed as a multistage game inwhich the principal first designs a mechanism, M, thatspecifies the terms of the contract. In particular, it sayswhat the salesperson would have to sell, and what he/shewould earn, both depending on what message he/shesends. After the mechanism has been designed, the sales-person accepts or rejects the contract. Suppose the sales-person accepts. Then, the information he/she has and theterms of the contract he/she chooses simultaneously senda costless message to the manager that will determinethe salesperson’s payoff. Thus, the principal chooses themechanism and the agent chooses the message (and theeffort), and the equilibrium in this game solves forthe optimal contract. An important result in the theoryof contracts is that the message can be restricted to bethe information the agent has, and an optimal contracthas the property that the agent would not have an incentiveto report his/her information falsely. This is known as therevelation principle, because the optimal contractuncovers the information that the principal does nothave. To see the intuition behind the revelation principle,suppose in the salesperson compensation problem thatthere are two salespersons with differing abilities, high(H) and low (L). Consider the (optimal) mechanism thatoffers m(H) and m(L), m(H) 6¼ m(L), if salespersons re-port H and L, respectively. Suppose, contrary to the rev-elation principle, the mechanism induces H to lie, andto report L instead. Then, because the mechanism isoptimal, the manager would have lost nothing by makingm(H)¼m(L). This sort of reasoning can be establishedmore rigorously. This is useful because it can restrictour attention to a subset of all possible contracts. Inlight of the revelation principle, the problem of optimalcontracting can now be thought of as maximizingexpected utility of the principal subject to three con-straints: IR, IC, and truth telling.

Using the Game Theoretic Model

There are far too many interesting applications of gametheory to cover here, but three examples with managerialimplications will give the reader a feel for how the con-cepts developed in the preceding discussions can be usedin modeling.

Auctions

Auctions play a major role in many spheres of economicactivity, including e-commerce, government sale of com-munication frequencies, sale of state-owned assets toprivate firms, and, of course, art. Sellers are interested inmaximizing their revenue and so analysis of equilibrium

under different auction rules for particular situationswould be helpful. From a larger perspective, it wouldbe desirable that the auctioned object goes to the personwho values it most. Again, how this is affected by specificauction rules can be studied using game theory. Atits core, an auction involves competitive bidding, withbidders taking turns. Bidders in an auction differ intheir valuation for an object and their knowledge of thetrue value, so auctions can be viewed as games of asym-metric information. Here the Nash equilibrium strategiesare characterized in the simple English and Dutch auc-tions. Consider a single object being auctioned to Npotential bidders, indexed by i. Bidder i knows his/hervaluation vi, of the object. Assume that 0 � vi � V51,so that all bidders have a nonzero valuation and none hasan infinite valuation. This type of auction is called a privatevalue auction because each bidder knows with certaintyhis/her value, but not others’ values. [In contrast, an auc-tion in which all bidders have the same (common) valuefor the object, but each has an error-prone estimate of thecommon value, is usually called a common value auction.In this case, knowing others’ estimates can help refine anindividual bidder’s estimate. The following discussionrelates only to private value auctions.]

In a Dutch auction, the auctioneer starts off at a highprice and decreases the price in small steps until a biddersteps in. The bidder wins at that price, but no information isgenerated about the valuations of the other potential bid-ders, except that they are lower than the winning bid. Be-cause the bidder pays the highest price, and nothing islearned during the auction, a Dutch auction is equivalentto a first-price sealed-bid auction. In an English auction, theauctioneer cries out bids in incremental steps, and biddersstay in or opt out. When a single bidder remains, he/she winsthe auction, and must pay the last bid at which the secondhighest bidder stayed in. During the course of the auction,information on the valuation of the other bidders becomesavailable from the price at which each dropped out, whichcould be useful, especially in common value auctions. Ina private value auction, because nothing is learned duringthe auction, an English auction is equivalent to a second-price sealed-bid auction. Seminal work by William Vickreyin 1961 showed that both first-price and second-price auc-tions in a private value auctions yield the same expectedrevenue to the seller.

Second-Price AuctionIn a second-price sealed-bid auction, the highest bidderwins but pays the second highest bid. LetBi denote i’s bid.The payoff to bidder i is then

pi Bi,B�ið Þ ¼vi �maxðB�iÞ if Bi 4maxðB�iÞ,0 else:

�

ð4Þ


The possibility of bids being tied is ignored. It is easy tosee that a Nash equilibrium to this game is for each bidderto bid Bi¼ vi. In the event that the bidder loses, thisstrategy is not dominated. In the event that the bidderwins, given other bidders’ strategies, this is a bestresponse. Thus, this strategy of bidding true valuation isa weakly dominant strategy, resulting in a Nash equilib-rium because all players are at their best response.

The seller is interested in the equilibrium revenue tothe seller, not the bids alone, and so would like to knowwhat the revenues are when using a second-price rule.Revenue would depend on the distribution of values overbidders. LetF(v) denote the cumulative distribution func-tion of the vi values. In other words, each bidder’s value isa draw from this distribution. Assume that the draws areindependent, and assume that F is differentiable and thevalues are in a closed interval. Given the Nash equilibriumstrategies, a bidder with valuation vi wins if all othershave a valuation less than vi, and this occurs, say, withprobability P(vi). Of course,P depends onF. KnowingF, itis also possible to calculate the expected value of thesecond highest bid as the maximum of (N� 1) indepen-dent draws from F conditioned on the maximum beingless than vi. Denote this by M(vi). Then, clearly the rev-enue to the seller conditioned on vi is P(vi)M(vi). Theexpected revenue can be obtained by unconditioningon vi. In other words, expected revenue to the seller issimply

RP(v)M(v) dF(v).

First-Price AuctionIn a first-price auction, the Nash equilibrium strategiesare slightly more complicated. Assume that bidders maxi-mize their expected payoff. Suppose all bidders followa strategy s(vi), with s monotonically increasing in vi.Then, s : vi!Bi. It can be shown that a Nash equilibriumin this case is s(vi)¼M(vi)5 vi. Thus, in a first-price auc-tion, all bidders bid less than their true value. But, therevenue to the seller, computed, as before, by first con-ditioning on vi and then taking expectations, is identical tothat in the second-price auction. This is Vickrey’s result.Thus, from the seller’s point of view, the two types ofauctions produce the same outcome. In both cases, theobject goes to the person who values it most. The equi-valence outcome under different auction rules does notalways obtain. Indeed, many features of auctions, such asthe use of reserve prices, auctions of multiple objects,common value auctions, and so on, would be part ofa model, depending on the situation. Since the time ofVickrey’s work, many game theoretic models have beendeveloped to incorporate these features.

Government Procurement

How might the government design a contract for defensepurchases? This is game of asymmetric information

because the (supplying) firm is likely to know moreabout its cost of supplying to the government than theknows about prices. Suppose the firm is one of two types,H and L, with abilities aH and aL, aH > aL, where cost ofsupplying the defense item is

ci ¼ c0 � ai � ei, i ¼ H,L, ð5Þ

where ei is the effort expended by firm i to control costs.The cost of the effort for the firm is V(e), V0, V00 4 0. Ofcourse, the government cannot observe the effort levelof a firm, but can observe the final cost. The governmentoffers a contract that pays t(c) if the firm’s reported costis c. The firm’s objective is to maximize

pi ¼ t cið Þ�V eið Þ: ð6Þ

The government cares about the cost, the social benefitB of the defense item, and the firm’s profits. So it wantsto maximize

pG ¼ B� c �V eið Þ� l cþ t cið Þ½ �: ð7Þ

The last term represents the social cost of taxes thatmust be used to pay the firm. Although the governmentdoes not know which type the firm is, it is assumed toknow the probability that the firm is type Hor L, and it iscommon knowledge. This problem can be seen as one ofmechanism design, with the government specifyinga payment s(H) and s(L), and associated costs s(H) ands(L), corresponding to messages H and L. For realiza-tions of other costs, the payment would be zero.

The solution to this problem yields a surprisingly sim-ple incentive contract. The contract reimburses a high-cost firm for its effort e, but this effort is lower than whatwould be optimal if there were no information asymmetryand the government knew that it was dealing with a high-cost firm. On the other hand, with a low-cost firm, thegovernment reimburses the cost of effort plus an addi-tional amount that elicits truth telling by the low-costfirm. The effort level chosen by the low-cost firm is exactlywhat it would have been with no asymmetric information.When there are many types of firms, rather than two, andcost observation is noisy, this result generalizes to a con-tract in which a firm receives a fixed payment anda variable payment that reimburses a fraction of the ob-served cost. The fixed payment is higher for a lower costfirm whereas a higher fraction of the cost is reimbursedfor a high-cost firm.

Retail Advertising and Pricing Practice

Supermarkets carry many products, and must decidewhich products they should price relatively low to inducecustomers to shop at their store, and also whether they


should advertise the price of such products. A gametheoretic model can help answer such questions. Con-sider two retailers, indexed by j, A and B located at theends of a straight line. Suppose that each carries two goods1 and 2, indexed by i, and the price of good i at store j is Pij.Consumers are willing to pay up to $R for either good.They are located on the lineAB, are uniformly distributed,and are assumed to incur a cost to visit a store proportionalto the distance of their location from the store. How mightthe competing stores price the products?

Peter Diamond has argued that, once at a store,a consumer would find it cheaper to buy at that store,even if the other store had a slightly higher price, becauseof the transportation cost. This would allow each store toprice a little above the other, eventually to the monopolyprice, in this case, R. This counterintuitive result ofmonopoly prices under competition has come to beknown as the Diamond paradox. The question is howthis would be affected if firms carrying multiple productscould advertise prices of only a subset of products? Thissortof question is relevant to supermarkets thatcarry manyproducts but can advertise only a few. This can be modeledas a two-stage game in which firms advertise the price ofone item (1 or 2) in the first stage, and then, after knowingthe advertising strategies, set the price of the other item inthe second stage. Of course, consumers would know theprices of advertised goods, but not the unadvertised ones.A Nash equilibrium to this game takes the following form:

Stage 1: Each firm advertises either item with prob-ability 0.5, and chooses a price Pa�5R for it.

Stage 2: If both stores advertised the same item (1 or 2)in stage 1, they set the price of the unadvertised good atR.If they advertised different items, they set the price ofthe unadvertised good at Pa� þ c5R, where c is the incre-mental cost for a consumer located at the midpoint ofroute AB to visit an additional store.

In this equilibrium, consumers find it optimal to buy bothitems at the nearest store, even when the stores advertisedifferent items, because, once they are at this store, itwould not pay them to take advantage of a lower priceon one of the items at the other store. When both storesadvertise the same item, it is clearly optimal for consumersto shop at the nearest store.

The mixed-strategy equilibrium is consistent with ca-sual observation of stores advertising both the same anddifferent products on occasion, stores offering unadver-tised price specials, and consumers content to make storechoices without knowing all prices. It also shows that atleast one item would always be below monopoly price R,while the expected price of the other item is also below R.The first effect is a consequence of permitting advertising,the second is a consequence of rational expectations onthe part of consumers. Thus, a game theoretic model canprovide insights into pricing practice.

Adapting the Game TheoreticModel

Repeated Games

An issue that has received a great deal of attention isthe effect of players repeatedly playing a game such asthe prisoners’ dilemma. If played only once, the equilib-rium is for both players to play their dominant strategy andto get a lower utility than what they would get with co-operation, but it can be conjectured that they could seek,and even obtain, some sort of cooperation if they encoun-tered each other repeatedly, in turn leading to an outcomethat is not a Nash equilibrium in a single play of the game.This line of thinking goes at least as far back as 1957, tothe work of R. Duncan Luce and Howard Raiffa, whoargued that ‘‘we feel that in most cases [of repetitionsof Prisoners’ Dilemma] an unarticulated collusionbetween the players will develop . . . this arises from theknowledge that . . . reprisals are possible.’’ However, forsome games, including the prisoners’ dilemma, it canbe shown that finite repetitions do not change the out-come, leading to what has come to be known as the chain-store paradox. Infinite repetitions are another matter,however. In this case, it is possible, under certain condi-tions, to support many payoff outcomes as a subgameperfect equilibrium, even if the outcome would not bea Nash equilibrium in a one-shot play of the game. Define,for each player, his/her security level or minmax value,mis, as follows:

mis ¼ mina� i

½maxai

pi ai, a� ið Þ�: ð8Þ

An individually rational payoff for player i is a payoff thatis not less than mis. Let players discount future payoffsby a factor d, 05d5 1. Let G(d) denote an infiniterepetition of a game G, with discount factor d. Then, theresult, called a folk theorem, holds for infinitely repeatedgames.

Folk TheoremAny vector of individually rational payoffs in a gameG canbe supported as a subgame perfect Nash equilibrium in aninfinitely repeated version of G for a sufficiently large d.The folk theorem actually requires an additional technicalcondition that the set of payoffs that strictly pareto-dominate the individually rational payoffs be of dimensionN. It is best to think of a desired level of payoffs to supportin equilibrium as a desired outcome. When a player de-viates from the desired outcome, the other players wouldpunish him/her. Now, the intuitive condition that playersshould not discount the future too much allows players toinflict sufficient punishment to any deviant behavior bya player. This is the reprisal contemplated by Luce andRaiffa. The technical condition allows players to punish


a deviant player without punishing themselves. No pun-ishment can lead to a player receiving a payoff lower thanhis/her minmax value. One of the consequences of the folktheorem is that sub-game perfect Nash equilibriumadmits too many possible outcomes, calling into questionthe usefulness of the equilibrium concept as a modelingdevice. The challenge then is to model a situation carefullyabout what players can observe and how they can imple-ment punishments. The main contribution of results suchas the folk theorem is the reasonable way it offers tocapture tacit understanding among players who interactrepeatedly.

Experiments and Games

There has been a good bit of effort to see how good a modelof human behavior game theory really is. Most of this workhas consisted of laboratory experiments using human par-ticipants and creating environments that would corre-spond to typical game theoretic models of competitiveinteraction, whether it be auctions or price determination.In one study, it was reported that some predictions ofgame theory describe behavior well. In particular,games with unique Nash equilibrium are replicatedwell in the lab. On the other hand, in the presence ofmultiple equilibria, theoretical procedures to refinethem do not find support in the lab. One of the conceptsemphasized here is the subgame perfect equilibrium.However, participants in an experiment may have diffi-culty in implementing such a complicated computationalprocedure. In contrast, the concept of fairness seems toaffect experimental outcomes, as do institutional struc-tures (for example, the number of players).

An issue in real-world competition is how players learn.In game theoretic models, the usual way to model learningis to use some sort of Bayesian updating. This does notalways accord well with experimental evidence. The ca-pacity of individuals to learn in the course of auctionsappears to be limited. Similar findings have been foundin a large number of experiments. These considerationshave led researchers to find ways of modeling learning,resulting in the growing area of evolutionary game theory.The goal of these efforts is to find a good way to incor-porate learning in a game theoretic model. Some scholarswho have explored this and related issues say there is littledoubt that game theory has much to offer as a model, andalternatives are not nearly as attractive. Even so, onequestion remains: what is the best way to use andadapt game theory as a model of rational players?

Acknowledgments

I thank Professors Nanda Kumar, Uday Rajan, andMiguel Vilas-Boas for their suggestions on early draftsof this article.

See Also the Following Articles

Gambling Studies � Quantitative Analysis, Economics

Further Reading

Abreu, D. (19880). Towards a theory of discounted repeatedgames. Econometrica 56, 383�396.

Ackerlof, G. (1970). The market for lemons: QualityUncertainty and the market mechanism. Q. J. Econ. 84,488�500.

Aumann, R., and Hart, S. (1992). Handbook of Game Theorywith Economic Applications. North Holland, New York.

Benoit, J.-P., and Krishna, V. (1985). Finitely repeated games.Econometrica 17, 317�320.

Binmore, K. (1990). Essays on the Foundations of GameTheory. Basil Blackwell Ltd., Oxford.

Davis, D. D., and Holt, C. A. (1993). Experimental Economics.Princeton University Press, Princeton, NJ.

Diamond, P. A. (1971). A model of price adjustment. J. Econ.Theory 3, 156�168.

Fudenberg, D., and Levine, D. K. (1998). The Theory ofLearning in Games. MIT Press, Cambridge, MA.

Fudenberg, D., and Maskin, E. (1986). The folk theorem inrepeated games with discounting or with incompleteinformation. Econometrica 54, 533�554.

Fudenberg, D., and Tirole, J. (1991). Game Theory. MITPress, Cambridge, MA.

Harsanyi, J., and Selten, R. (1988). A General Theoryof Equilibrium Selection in Games. MIT Press,Cambridge, MA.

Kagel, J. H., and Roth, A. E. (1995). The Handbook ofExperimental Economics. Princeton University Press, Prin-ceton, NJ.

Krishna, V. (2002). Auction Theory. Academic Press, SanDiego, CA.

Lal, R., and Matutes, C. (1994). Retail pricing and advertisingstrategies. J. Bus. 67(3), 345�370.

Luce, R. D., and Raiffa, H. (1957). Games and Decisions:Introduction and Critical Survey. Wiley, New York.

McAfee, R. P., and McMillan, J. (1987). Auctions and bidding.J. Econ. Lit. 25, 699�754.

Milgrom, P., and Roberts, J. (1986). Price and advertisingsignals of product quality. J. Pol. Econ. 94, 796�821.

Milgrom, P., and Weber, R. (1982). A theory of auctions andcompetitive bidding. Econometrica 50, 1089�1122.

Nash, J. (1950). Equilibrium points in n-person games. Proc.Natl. Acad. Sci. U.S.A. 36, 48�49.

Rao, R. C., and Syam, N. (2000). Equilibrium price commu-nication and unadvertised specials by competing super-markets. Market. Sci. 20(1), 66�81.

Ross, S. (1977). The determination of financial structure: Theincentive-signalling approach. Bell J. Econ. 8, 23�40.

Selten, R. (1965). Spieltheoretische Behandlung eines Oligo-polmodells mit Nachfragetragheit. Z. Ges. Staatswiss. 121,301�324, 667�689.

Selten, R. (1978). The chain-store paradox. Theory Decis. 9,127�159.

Smith, V. L. (1989). Theory, experiment and economics.J. Econ. Perspect. 3(1), 151�169.


Spence, A. M. (1974). Market Signalling: InformationalTransfer in Hiring and Related Processes. Harvard Uni-versity Press, Cambridge.

Stahl, D. O. (1989). Oligopolistic pricing with sequentialconsumer search. Am. Econ. Sci. Rev. 14(4), 700�712.

Vickrey, W. (1961). Counterspeculation, auctions, and compe-titive sealed tenders. J. Finance 16, 8�37.

Wilson, R. (1993). Strategic analysis of auctions. Handbook ofGame Theory (R. Aumann and S. Hart, eds.). Amsterdam,North Holland.


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