Game Theory. What is 'Game Theory'Game theory is the study of human conflict and cooperation within a competitive situation. In some respects, game theory is the science of strategy, or at least the optimal decision- making of independent and competing actors in a strategic setting. The key pioneers of game theory were mathematicians John von Neumann and John Nash, as well as economist Oskar Morgenstern. The term “game” can be misleading. Even though game theory applies to recreational games, the concept of “game” simply means any interactive situation in which independent actors share more- or- less formal rules and consequences. The formal application of game theory requires knowledge of the following details: the identity of independent actors, their preferences, what they know, which strategic acts they are allowed to make, and how each decision influences the outcome of the game. Depending on the model, various other requirements or assumptions may be necessary. Finally, each independent actor is assumed to be rational. Game theory has a wide range of applications, including psychology, evolutionary biology, war, politics, economics and business. ![]()
Despite its many advances, game theory is still a young and developing science. Impact on Economics and Business. Game theory brought about a revolution in economics by addressing crucial problems in prior mathematical economic models. For instance, neoclassical economics struggled to understand entrepreneurial anticipation and couldn't handle imperfect competition. Game theory turned attention away from steady- state equilibrium and toward market process. In game theory, every decision- maker must anticipate the reaction of those affected by the decision. ![]() In business, this means economic agents must anticipate the reactions of rivals, employees, customers and investors. An Example of Game Theory. Suppose executives in charge of Apple i. OS and Google Android are deciding whether or not to collude and exert duopolistic power over the market for smartphone operating software. Each firm knows that if they work together and do not cheat each other, they will be able to restrict output and raise prices, thereby enjoying above- normal profits. A simple application of the prisoner’s dilemma shows Apple and Google cannot maintain a stable equilibrium while colluding together, even under the unrealistic assumption that no other market competitors exist or could exist. Consider the four possible scenarios: 1. Both Apple and Google sell the agreed- upon amount, do not cheat, and enjoy above- normal profits. Apple only sells the agreed- upon amount of operating software, but Google sells the quantity at which it receives maximal net return (perhaps through secret rebates or setting up a shadow subsidiary). Google realizes even greater profits by discretely offering goods at sub- duopoly prices, and Apple loses market share. Google doesn’t cheat, but Apple does. Apple realizes even greater profits by cheating, and Google loses market share. An oligopoly (from Ancient Greek An illustrated tutorial on how game theory applies to pricing decisions by firms in an oligopoly, how a firm can use a dominant strategy to produce its best results. Game theory is "the study of mathematical models of conflict and cooperation between intelligent rational decision-makers." Game theory is mainly used in economics. Both Apple and Google compete normally and realize normal profits. Whether or not Google cheats, Apple is better off cheating, and vice versa. The same logic holds true whether discussing individual brokers, advisors, salesmen or entire firms. Oligopoly. Defining and measuring. An oligopoly is a. When a market is shared between a few firms, it is said. Although only a few firms dominate, it is. For example, major. British Airways (BA) and. Air France operate their routes. Concentration ratios. Oligopolies may be identified using. When there is a high. Example of a hypothetical concentration ratio. The following. are the annual sales, in . The H- H index is found by adding together the squared values. For example, if three. X2 + Y2 + Z2; where X, Y and Z. If the index is below 1. Mergers between oligopolists increase concentration. A firm operating in a market with. For example, if a. Texaco wishes to increase its market share by. Shell and BP, may reduce their price in retaliation. An understanding of. Dilemma helps appreciate the concept of interdependence. Strategy. Strategy is extremely important to firms that are. Because firms cannot act independently, they must. In other words, they need to plan, and work out a range of possible options based on how they think. Oligopolists have to make critical strategic. Whether to compete with rivals, or collude with. Whether to raise or lower price, or keep price. Whether to be the first firm to implement a new. The advantages. of . Sometimes it pays to go first because a firm. Barriers to entry. Oligopolies and monopolies. These hurdles are called barriers to entry and the incumbent. Natural entry barriers include: Economies of large scale production. If a market has significant. Ownership or control of a key scarce resource Owning scarce resources that other firms would like. High set- up costs High set- up costs deter initial market entry. In order to compete, new entrants will have to. This deters entry, and is widely found in. Artificial barriers include: Predatory pricing Predatory pricing occurs when a firm deliberately. Limit pricing Limit pricing means the incumbent firm sets a low. This signals to. potential entrants that profits are impossible to make. Superior knowledge. An incumbent may, over time, have built up a. This superior knowledge can deter entrants into the. Predatory acquisition. Predatory acquisition involves taking- over a. As with other deliberate barriers. Markets Authority (CMA), may prevent this because it is likely to reduce. Advertising. Advertising is another. A strong brand. A strong brand creates loyalty, . For example. contracts between suppliers and retailers can exclude other. Vertical integration. Vertical integration can . If colluding. participants act like a monopoly and can enjoy the benefits of higher. Types of collusion. Overt. Overt collusion occurs when there is no attempt to. Association of Petrol Retailers. Covert. Covert collusion occurs when firms try to hide the. Tacit. Tacit collusion arises when firms act together. For example, it may be accepted that a. All firms may . If firms do collude, and their. In many cases, tacit collusion is. When competing, oligopolists prefer non- price. A price reduction may achieve strategic benefits. This leads to little or no gain, but can lead to. Hence, a far more beneficial strategy may be. Pricing strategies of oligopolies. Oligopolies may pursue the following pricing. Oligopolists may use predatory pricing to. This means keeping price artificially low, and. They may also operate a limit- pricing strategy. Oligopolists may collude with rivals and raise price. Cost- plus pricing is a straightforward pricing. Cost- plus pricing is also called rule of thumb pricing. There are different versions of cost- pus. Cost- plus pricing is very useful for firms that. Hence, it can be regarded as a. Cost- plus pricing is also common in. However, there is a risk with such a rigid. Cost- plus pricing can also be explained through. This. takes some of the risk out of pricing decisions, given that all. This could be considered a form of. Non- price strategies. Non- price competition is the favoured strategy for. Trying to improve quality and after sales. Spending on advertising, sponsorship and. The UK's football. Premiership has long. Barclays Bank and. Carling. Sales promotion, such as. BOGOF), is associated with the large. Loyalty schemes, which are common in the. Sainsbury’s. Nectar Card and. Tesco’s. Club Card. Each strategy can be evaluated in terms of: How successful is it likely to be? Will rivals be able to copy the strategy? Will the firms get a 1st - mover advantage? How expensive is it to introduce the strategy? A strategy that. takes five years to generate a pay- off may be rejected in favour of. Price stickiness. The theory of oligopoly suggests that, once a price. This is largely because firms cannot pursue. For example, if an airline raises the. London to New York, rivals will not follow. Rivals have no need to follow suit because. However, if the airline lowers its price, rivals would be. Again, the. airline will lose sales revenue and market share. The demand curve is. Kinked demand curve. The reaction of rivals to a price change depends on. The elasticity of demand, and hence. The demand. curve will be kinked, at the current price. Even when there is a large rise in marginal cost. At price P, and output Q, revenue will be maximised. Maximising profits. If marginal revenue and marginal costs are added it is possible to show. P. Profits will always be maximised when MC = MR, and so long as MC cuts. MR in its vertical portion, then profit maximisation is still at P. Even when MC moves out of the vertical portion. A game theory approach to price stickiness Pricing strategies can also be looked at in terms of. There are three. possible price strategies, with different pay- offs and risks: Raise price. Lower price. Keep price constant. The choice of strategy will depend upon the. Raising price. or lowering price could lead to a beneficial pay- off, but both. In. short, changing price is too risky to undertake. Therefore, although keeping price constant will not lead to. The Prisoner’s Dilemma Game theory also predicts that: There is a tendency for cartels to form because. Co- operation reduces the. While cartels are . Cartels are designed to protect the interests of. Higher prices or hidden prices, such as the. Lower output Restricted choice or other limiting conditions. A classic game called the. Dilemma is often used to demonstrate the interdependence of. Examples of Oligopoly. Oligopolies are common in the airline industry. By making decisions more complex - . Firms can be prevented from entering a market. There is a potential loss of economic welfare. Oligopolists may be allocatively and. Oligopolies tend to be both allocatively and. At profit maximising equilibrium, P, prce is. MC, and output, Q, is less than the productively efficient output. Q1, at point A. The advantages of oligopolies. However, oligopolies may provide the following. Oligopolies may adopt a highly competitive. Even though. there are a few firms, making the market uncompetitive, their behaviour may be highly competitive. Oligopolists may be dynamically. The super- normal profits they generate may be used to. Price stability may bring advantages to. See Transfer. pricing. See Browser market shares. Game Theory . Either she continues according to plan, hoping that her partner does likewise (because she cannot bag a deer on her own), and together they catch the deer; or she goes for a hare instead, securing a prey that does not require her partner’s cooperation, and thus abandoning the common plan. Each hunter prefers a deer shared between them to a hare for herself alone. But if she decides to hunt for deer, she faces the possibility that her partner abandons her, leaving her without deer or hare. So, what should she do? And, what will she do? Situations like this, in which the outcome of an agent’s action depends on the actions of all the other agents involved, are called interactive. Two people playing chess is the archetypical example of an interactive situation, but so are elections, wage bargaining, market transactions, the arms race, international negotiations, and many more. Game theory studies these interactive situations. Its fundamental idea is that an agent in an interactive decision should and does take into account the deliberations of her opponents, who, in turn, take into account her deliberations. A rational agent in an interactive situation should therefore not ask: “What can I do, given what is likely to happen?” but rather: “What can I do in response to what they do, given that they have a belief about what I will do?” Based on this perspective, game theory recommends rational choices for these situations, and predicts agents’ behavior in them. This article presents the basic tenets of game theory in a non- formal way. It then discusses two broad philosophical issues arising from the theory. First, whether the rationality concept employed by the theory is justifiable – whether it is intuitively rational to choose as the theory prescribes. Second, whether the theory can in principle be a good predictive theory of human behavior – whether it has empirical content, whether it is testable and whether there are good reasons to believe that it is true or false. Table of Contents. Sketch of the Theory. Static Games. Dynamic Games. The Architecture of Game Theory. Game Theory as a Theory of Rationality. Sufficient Epistemic Conditions for Solution Concepts. Nash Equilibrium in One- Shot Games. Nash Equilibrium in Repeated Games. Backward Induction. Paradoxes of Rationality. Bounded Rationality in Game Players. Game Theory as a Predictive Theory. The Evolutive Interpretation. The Problem of Alternative Descriptions. Testing Game Theory Conclusion. References and Further Reading 1. Sketch of the Theory. Game theory belongs to a family of theories often subsumed under the umbrella term Rational Choice Theory. All these theories (in particular, decision theory, game theory and social choice theory) discuss conditions under which agents’ actions, or at least their decision to act, can be said to be rational. Depending on how these conditions are interpreted, Rational Choice theory may have a positive or a normative function: it may contribute to the prediction and explanation of agent behavior, or it may contribute to advising agents what they should do. Many of the purported functions of Rational Choice theory are controversial; as a part of it, game theory is affected by these controversies, in particular its usefulness for the social sciences. I will address some of these general issues in Section 3. However, game theory faces its own philosophical problems, and these will be the focus of this article. Decision theory, as well as game theory, assesses the rationality of decisions in the light of preferences over outcomes and beliefs about the likelihood of these outcomes to appear. The basic difference between the two lies in the way they view the likelihood of outcomes. Decision theory treats all outcomes as exogenous events, . Game theory, in contrast, focuses on those situations in which outcomes are determined by interactions of deliberating agents. It proposes that agents take outcomes as determined by other agents’ reasoning, and that agent therefore assess the likelihood of an outcome by trying to figure out how the other agents they interact with will reason. The likelihoods of outcomes therefore becomes “endogenous” in the sense that players take their opponents’ payoffs and rationality into account when figuring out the consequences of their strategies. We are familiar with such reasoning from card and board games. When playing poker or chess, one must take one’s opponent’s reasoning into account in order to be successful. The player who foresees her opponent’s optimal reaction to her own move will be much more successful that the player who simply assumes that her opponent will make a certain move with a certain probability. Theoretical reflection about such parlor games are at the basis of game theory – for example, James Waldegrave’s discussion of the French card game Le Her in 1. Neumann’s treatment 'Zur Theorie der Gesellschaftsspiele' (. A player’s pure strategy specifies her choice for each time she has to choose in the game (which may be more than once). Players have to have at least two strategies to choose between, otherwise the game would be trivial. All players of a game together determine a consequence. Each chooses a specific strategy, and their combination (called strategy profiles) yields a specific consequence. The consequence of a strategy profile can be a material prize – for example money – but it can also be any other relevant event, like being the winner, or feeling guilt. Game theory is really only interested in the players’ evaluations of this consequence, which are specified in each players’ so- called payoff or utility function. The part of the theory that deals with situations in which players’ choice of strategies cannot be enforced is called the theory of non- cooperative games. Cooperative game theory, in contrast, allows for pre- play agreements to be made binding (e. This article will not discuss cooperative game theory. More specifically, it will focus – for reasons of simplicity – on non- cooperative games with two players, finite strategy sets and precisely known payoff functions. Game theory uses two means to represent games formally: strategic form and extensive form. Commonly (though not necessarily!), these two methods of representation are associated with two different kinds of games. Extensive form games represent dynamic games, where players choose their actions in a determined temporal order. Strategic form games represent static games, where players choose their actions simultaneously. Static Games. Static two- person games can be represented by m- by- n matrices, with m rows and n columns corresponding to the players’ strategies, and the entries in the squares representing the payoffs for each player for the pair of strategies (row, column) determining the square in question. As an example, figure 1 is a possible representation of the stag- hunt scenario described in the introduction. Col's Choice. C1. C2. Row's. Choice. R1. 2,2. 0,1. R2. Figure 1: The stag hunt. The 2- by- 2 matrix of figure 1 determines two players, Row and Col, who each have two pure strategies: R1 and C1 (go deer hunting) and R2 and C2 (go hare hunting). Combining the players’ respective strategies yields four different pure strategy profiles, each associated with a consequence relevant for both players: (R1,C1) leads to them catching a deer, (R2,C1) leaves Row with a hare and Col with nothing, (R2,C2) gets each a hare and (R1,C2) leaves Row empty- handed and Col with a hare. Both players evaluate these consequences of each profile. Put informally, players rank consequences as . In the stag- hunt scenario, players have the following ranking: Row. Col. 1. Utility functions are used to represent players’ evaluations of consequences in games (for more on preferences and utility functions, see Gr. Convention has it that the first number represents Row’s evaluation, while the second number represents Col’s evaluation. It is now easy to see that the numbers of the game in figure 1 represent the ranking of figure 2. Note, however, that the matrix of figure 1 is not the only way to represent the stag- hunt game. Because the utilities only represent rankings, there are many ways how one can represent the ranking of figure 2. For example, the games in figures 3a- c are identical to the game in figure 1. C1. C2. R1- 5,- 5- 7,- 6. R2- 6,- 7- 6,- 6. Figure 3a: 3rd version of the stag hunt. C1. C2. R1. 10. 0,1. R2. 99,1. 99,9. 9Figure 3b: 2nd version of the stag hunt. C1. C2. R1- 5,1. 00- 7,- 9. R2- 6,1- 6,9. 9Figure 3c: 1st version of the stag hunt. In figure 3a, all numbers are negative, but they retain the same ranking of consequences. And similarly in figure 3b, only that here the proportional relations between the numbers (which don’t matter) are different. This should also make clear that utility numbers only express a ranking for one and the same player, and do not allow to compare different players’ evaluations. In figure 3c, although the numbers are very different for the two players, they retain the same ranking as in figure 1. Comparing, say, Row’s evaluation of (R1,C1) with Col’s evaluation of (R1,C1) simply does not have any meaning. Note that in the stag- hunt game, agents do not gain if others lose. Everybody is better off hunting deer, and losses arise from lack of coordination. Games with this property are therefore called coordination games. They stand in stark contrast to games in which one player’s gain is the other player’s loss. Most social games are of this sort: in chess, for example, the idea of coordination is wholly misplaced. Such games are called zero- sum games. They were the first games to be treated theoretically, and the pioneering work of game theory, von Neumann and Morgenstern’s (1. The Theory of Games and Economic Behavior concentrates solely on them.
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