Fiona Hayes
Senior Sophister

Game theory is best described as cold blooded rational interactive decision making.[4] It is the extension of individual rational decision making to the behaviour of rational decision makers whose decisions affect each other. In recent decades much progress has been made in applying game theoretic models to a wide range of economic problems. Indeed, Varian claims that most economic behaviour can be viewed as special cases of game theory. Game theory is divided into two branches, cooperative and non-cooperative. In non-cooperative game theory, individuals cannot make binding agreements and the unit of analysis is the individual who is concerned with doing as well as possible for himself, subject to clearly defined rules and possibilities. In cooperative game theory, binding agreements are allowed and the unit of analysis is the group or coalition. This discussion will concern itself exclusively with non-cooperative game theory. Furthermore, I will avoid complex mathematical equations and detailed analysis of the fundamentals of the theory, choosing instead to outline the most important concepts in simple terms and to see how the theory applies to specific economic examples.

One preliminary issue needs to be addressed in order to approach the discussion in context: by what standard are we to judge the usefulness of a theory? Kreps (1990) suggests that a useful theory is one which helps us to understand or predict behaviour in concrete economic situations. Hence studying the interactions of ideally rational people should aim to help our understanding of the behaviour of real individuals in real economic situations.

Formulation of the Game

Von Neumann and Morgenstern (1944) first recognised explicitly that economic agents must take into account the interactive nature of situations when making their decisions. Their methodology consisted of taking an economic problem, formulating it as a game, finding the game theoretic solution, and translating the solution back into economic terms.

So how do we formulate an economic problem as a game? The essential elements of a game are players, actions, information, strategies, payoffs and equilibria. The sets of players, strategies and payoffs combine to give us the rules of the game. There are two models the strategic and extensive forms. Indeed, the former can be viewed as a summarized description of the latter. Two equilibrium concepts commonly used are dominant strategy equilibrium and Nash equilibrium. A dominant strategy must be the best response to all possible strategy combinations by other players. A dominant strategy equilibrium is a strategy combination consisting of all players dominant strategies. It can be arrived at by iterative deletion of dominated strategies, i.e. eliminating all actions that the players will not choose. This requires common knowledge of rationality. A Nash equilibrium requires that si*[5] be the best response to a particular strategy combination by other players. The idea here is that no single player has the incentive to deviate. We can have equilibria in pure or mixed strategies. Nash equilibrium is the single game theoretic concept most frequently applied in economic examples. In all the examples which follow we will concern ourselves with the search for a Nash equilibrium (or some refinement thereof).

The Games

Consider this first example, known as the Prisoners Dilemma. We have two prisoners in separate cells, faced with the dilemma of whether or not to inform. Each player has defect as best response to any action by the other. The only Nash equilibrium is (defect,defect), despite the fact that both would be better off if they were to cooperate, because neither player acting unilaterally has the incentive to deviate. Note this is also a dominant strategy equilibrium (DSE).

The Prisoners Dilemma

                       Player 1           
                  Cooperate      Defect    
                  (Advertise)    (Do Not)   
        Cooperate    (-1,-1)     (-5,0)    
Player  (Advertise)                               
         Defect      (0,-5)      (-4,-4)   
        (Do Not)                        

This general sort of situation arises in many contexts in economics. For example, where we have two firms selling the same product and deciding whether to advertise or not. The Nash and dominant strategy equilibrium is (do not, do not).

Next let us turn to a Battle of the Sexes game. This game is representative of many situations in which two or more parties seek to coordinate their actions, although they have different preferences concerning how to coordinate. In this simple example, player 1 would prefer to go to a Chinese restaurant, while player 2 would prefer Italian, but both would prefer dining together to dining alone. We have two pure strategy Nash equilibria at (Ch,Ch) and at (It,It). We do not know which of these will be selected. Perhaps there is a focal point an equilibrium which for psychological and other reasons is particularly attractive. For example, the couple in question may have eaten Chinese food the previous night.

The Battle of the Sexes

                    Player 1          
                Chinese  Italian  
Player  Chinese  (6,3)    (0,0)    
        Italian  (0,0)    (3,6)    

There are many such coordination problems in macroeconomics. Kreps cites the example of two adjacent tax authorities who wish to coordinate on the tax system they employ in order to prevent taxpayers benefiting from any differences.

Application to Oligopoly Theory

One of the most successful applications of game theory has been to oligopoly theory. Oligopoly is concerned with market structures in which the actions/payoffs of the individual firms affect and are affected by the actions/payoffs of other firms. I will discuss three models of oligopoly, restricting our attention to duopoly for the sake of simplicity. We assume the firms are producing an homogeneous product.

Cournots Analysis

In Cournots (1838) model, firms choose quantities (ql,q2) from strategy space, S=(0,). Firms payoff functions are their respective profit functions:

where q = ql + q2 and p(q) is the inverse demand curve.

There is interdependence i.e. 1 depends not only on ql but also on q2. A Cournot Nash equilibrium occurs at a pair of output levels (ql*,q2*) such that neither firm could have obtained higher profit by having chosen some other output i.e. no player has incentive to deviate. Let us consider the simple example of a linear demand curve and constant marginal cost:

and, by symmetry:

These are the reaction curves for firm 1 and firm 2 respectively, and show the optimum reaction for each firm given how the other has reacted. Substituting, we obtain the result that:

So our equilibrium pair of output levels is

Stackelbergs Analysis

Stackelberg (1934) proposed a basic derivative of the Cournot model: the leader-follower duopoly, in which one firm, say firm 1 chooses q1 and, after that choice is communicated to firm 2, q2 is chosen. Firm 2 will always choose q2 according to q2 = f(ql) and this is known to firm 1 who maximizes q1. The output vector Q = (ql,q2) is a Stackelberg equilibrium with firm 1 as the leader and firm 2 as the follower if firm 2 maximizes profit subject to the constraint that firm 2 chooses according to his reaction function. Once again, neither firm can increase his profit by a unilateral decision to alter its output. Refering back to our simple example, the Stackelberg equilibrium is

We see that the profits of the leader will be higher than those of the follower because of first mover advantage. We also note that more is produced in this model than in Cournots model.

Bertrands Analysis

Bertrand (1883) realised that the focus on quantity competition was unrealistic. He proposed an alternative model whereby firms compete via price. Firms choose prices (p1, p2) from strategy space. The firms payoff functions are given by:

(Assuming a linear demand curve and constant marginal cost.)

The only Bertrand Nash equilibrium is where pl=p2=c and no firm can make a (higher) profit by altering its own price decision, i.e. even with only two firms we obtain competitive results. This is the Bertrand paradox: we know that firms do compete in prices and that they do make positive profits in the real world.

Returning once again to our simple example, we see that the Bertrand equilibrium outcome is the same as in the Cournot model. Despite the fact that the three models differ, they can all be seen as the application of the Nash equilibrium concept to games which differ with respect to the choice of strategic variables and the timing of moves. Cournot and Bertrand equilibria are the Nash equilibria of simultaneous move games where the strategic variables are quantities and prices respectively. The Stackelberg equilibrium is the subgame perfect (explained below) Nash equilibrium of a game where quantities are chosen, but the leader moves before the follower. We can conclude that, despite the fact that the same equilibrium concept is applied to all three models, the different outcomes suggest that oligopoly theory results are very sensitive to the details of the model.[6]

Dynamic Games

Thus far we have concentrated almost exclusively on Nash equilibria in their purest sense. The Stackelberg model hinted at further issues to be addressed. The first of these is that of time in games: what happens if we repeat a prisoners dilemma type game infinitely often? The Folk Theorem tells us that players acting non-cooperatively can attain the cooperative outcome via trigger strategies, under certain conditions. In this case, it is in the long run interest of prisoners to cooperate, but it is in their short run interest to defect; only the (Defect,Defect) outcome can occur in equilibrium, but if the discount rate is sufficiently high, we may get a cooperative outcome.

Repeated games model the psychological, informational side of ongoing relationships.[7] Phenomena like altruism, trust, revenge and punishment are all predicted by the theory. In repeated games, payoffs in each period are determined solely by actions in that period, yet strategies are a function of the entire history of the game up to that period. Dynamic games reflect the fact that current actions affect not only current payoffs, but also opportunities in the future - we learn from others, we also teach.

Refinements of Nash Equilibrium

                Accommodate   Fight    
Entrant  Enter     (-1,-1)   (-10,0)   
        Stay out   (0,-10)   (-8,-8)   

The entrant-incumbent example can be viewed as a two period game where the entrant makes his decision in the first period and the incumbent responds in the second period. This is our starting point for a brief look at some of the refinements of Nash equilibrium. If this is a once off game, (5,5) is a Nash equilibrium. So too is (0,10). But in the latter case the problem is that the threat of fight, if not enter, is not credible. Equilibria supported by incredible threats are not subgame perfect. (A subgame perfect Nash equilibrium must be the Nash equilibrium of every subgame within the game in question). Hence the only subgame perfect Nash equilibrium is (5,5). This concept can be applied to the problem of whether or not government plans are sustainable.

A related refinement is the concept of trembling hand perfection. Suppose there exists a small probability that players dont play their dominant strategies. A trembling hand perfect equilibrium must be robust against slight perturbations in strategy: (5,5) is trembling hand perfect; (0,10) is not. Issues surrounding this concept are similar to the problems of applying rational expectations models to study reforms and regime changes.[8]

No discussion of game theory is complete without reference to the crucial issue of information in games, i.e. what do players know about each other? There are four separate categories of information, but here we consider only incomplete information models i.e. where nature chooses a type for one player and this goes unobserved by at least one player. The solution concept we use is known as Bayesian Nash equilibrium, a simple variation of the concept we have used in all models to date. A specific category is signalling games. The classic example is Spences (1975) model of education as a signaling device in the job market. Other examples include price as a signal of quality in the goods market. Further examples of games with incomplete information include auctions (where parties hold proprietary information).

A Final Analysis

At this stage we have seen that the game theoretic approach can be applied to virtually all interactive decisions in economics. But what about the deficiencies? Firstly, game theory requires clearly defined rules of the game, and there is a tendency to take these rules too much for granted, without asking from where they come. Secondly, in many games we have multiple equilibria and no way to choose, e.g. Edgeworths bilateral bargaining problem in cooperative game theory. Thirdly, we have inefficient outcomes in many cases, because of payoff dependence. This can be seen as a type of externality. For example, in the Cournot duopoly model, both firms would do better to restrict output to half the monopoly output, but this is not a sustainable equilibrium as both have the incentive to cheat and produce more.

Despite these shortcomings, non-cooperative game theory has provided a unified and flexible language for analysing interactive decision-making in a wide variety of economic contexts. Often the results obtained are closely related to those from the more conventional approaches; in other cases, game theoretic models lead to new insights. Furthermore, game theoretic models enable us to identify similarity in superficially different situations, and to move insights from one context to another. In addition, game theory rightly stresses the importance of mechanics i.e. who moves when and with what information. However, we must keep a sense of proportion about the usefulness of the theory. The problems mentioned in the previous paragraph suggest that we must supplement game theory with considerations about experience and memory[9] that cannot be incorporated into the formal structure of the theory. In conclusion, I echo the words of Kreps who maintains that we should be happily dissatisfied overall - happy with the progress that has been made, but dissatisfied with our inadequate knowledge about how individuals actually behave in a complex and dynamic world.


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