Game theory is best described as cold blooded rational interactive decision making. 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.
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* 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).
Player 1 Cooperate Defect (Advertise) (Do Not) Cooperate (-1,-1) (-5,0) Player (Advertise) 2 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.
Player 1 Chinese Italian Player Chinese (6,3) (0,0) 2 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.
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
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.
(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.
Repeated games model the psychological, informational side of ongoing relationships. 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.
Incumbant 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.
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).
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 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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