Game Theory

Key Concepts

A game is any situation in which players (the participants) make strategic decisions—i.e., decisions that take into account each other’s actions and responses.

Payoffs are the value associated with a possible outcome.

A strategy is a rule or plan of action for playing the game.

Optimal strategy is the strategy that maximises a player’s expected payoff.

Cooperative game is a game in which participants can negotiate binding contracts that allow them to plan joint strategies.

Non-cooperative game is the game in which negotiation and enforcement of binding contracts are not possible.

An example of a cooperative game is the bargaining between a buyer and a seller over the price of a rug. If the rug costs $100 to produce and the buyer values the rug at $200, a cooperative solution to the game is possible: An agreement to sell the rug at any price between $101 and $199 will maximise the sum of the buyer’s consumer surplus and the seller’s profit, while making both parties better off.

An example of a non-cooperative game is a situation in which two competing firms take each other’s likely behaviour into account when independently setting their prices. Each firm knows that by undercutting its competitor, it can capture more market share. But it also knows that in doing so, it risks setting off a price war. Another non-cooperative game is the auction mentioned above: Each bidder must take the likely behaviour of the other bidders into account when determining an optimal bidding strategy.

Dominant Strategies

Dominant strategy is the strategy that is optimal no matter what an opponent does.

The following example illustrates this in a duopoly setting. Suppose Firms A and B sell competing products and are deciding whether to undertake advertising campaigns. Each firm will be affected by its competitor’s decision. The possible outcomes of the game are illustrated by the payoff matrix in Table below. Observe that if both firms advertise, Firm A will earn a profit of 10 and Firm B a profit of 5. If Firm A advertises and Firm B does not, Firm A will earn 15 and Firm B zero. The table also shows the outcomes for the other two possibilities.

What strategy should each firm choose? First consider Firm A. It should clearly advertise because no matter what firm B does, Firm A does best by advertising. If Firm B advertises, A earns a profit of 10 if it advertises but only 6 if it doesn’t. If B does not advertise, A earns 15 if it advertises but only 10 if it doesn’t. Thus advertising is a dominant strategy for Firm A. The same is true for Firm B: No matter what firm A does, Firm B does best by advertising. Therefore, assuming that both firms are rational, we know that the outcome for this game is that both firms will advertise. This outcome is easy to determine because both firms have dominant strategies. When every player has a dominant strategy, we call the outcome of the game an equilibrium in dominant strategies.

Unfortunately, not every game has a dominant strategy for each player. To see this, let’s change our advertising example slightly. The payoff matrix in Table 13.2 is the same as in Table 13.1 except for the bottom right-hand corner—if neither firm advertises, Firm B will again earn a profit of 2, but Firm A will earn a profit of 20. (Perhaps Firm A’s ads are expensive and largely designed to refute Firm B’s claims, so by not advertising, Firm A can reduce its expenses considerably.)

Now Firm A has no dominant strategy. Its optimal decision depends on what Firm B does. If Firm B advertises, Firm A does best by advertising; but if Firm B does not advertise, Firm A also does best by not advertising. Now suppose both firms must make their decisions at the same time. What should Firm A do?

To answer this, Firm A must put itself in Firm B’s shoes. What decision is best from Firm B’s point of view, and what is Firm B likely to do? The answer is clear: Firm B has a dominant strategy—advertise, no matter what Firm A does. (If Firm A advertises, B earns 5 by advertising and 0 by not advertising; if A doesn’t advertise, B earns 8 if it advertises and 2 if it doesn’t.) Therefore, Firm A can conclude that Firm B will advertise. This means that Firm A should advertise (and thereby earn 10 instead of 6). The logical outcome of the game is that both firms will advertise because Firm A is doing the best it can, given Firm B’s decision; and Firm B is doing the best it can, given Firm A’s decision. 

Nash equilibrium

Nash equilibrium is a set of strategies (or actions) such that each player is doing the best it can, given the actions of its opponents. Because each player has no incentive to deviate from its Nash strategy, the strategies are stable.

Dominant Strategies: I’m doing the best I can no matter what you do. You’re doing the best you can no matter what I do.

Nash Equilibrium: I’m doing the best I can, given what you are doing. You’re doing the best you can, given what I am doing.

Note: that a dominant strategy equilibrium is a special case of a Nash equilibrium.

In the advertising game of Table 13.2, there is a single Nash equilibrium—both firms advertise. In general, a game need not have a single Nash equilibrium. Sometimes there is no Nash equilibrium, and sometimes there are several (i.e., several sets of strategies are stable and self-enforcing). A few more examples will help to clarify this.

The Product choice Problem

Consider the following “product choice” problem. Two breakfast cereal companies face a market in which two new variations of cereal can be successfully introduced—provided that each variation is introduced by only one firm. There is a market for a new “crispy” cereal and a market for a new “sweet” cereal, but each firm has the resources to introduce only one new product. The payoff matrix for the two firms might look like the one in Table 13.3. 

In this game, each firm is indifferent about which product it produces—so long as it does not introduce the same product as its competitor. If coordination were possible, the firms would probably agree to divide the market. But what if the firms must behave non-cooperatively? Suppose that somehow—perhaps through a news release—Firm 1 indicates that it is about to introduce the sweet cereal, and that Firm 2 (after hearing this) announces its plan to introduce the crispy one. Given the action that it believes its opponent to be taking, neither firm has an incentive to deviate from its proposed action. If it takes the proposed action, its payoff is 10, but if it deviates—and its opponent’s action remains unchanged—its payoff will be 5. Therefore, the strategy set given by the bottom left-hand corner of the payoff matrix is stable and constitutes a Nash equilibrium: Given the strategy of its opponent, each firm is doing the best it can and has no incentive to deviate.

Note that the upper right-hand corner of the payoff matrix is also a Nash equilibrium, which might occur if Firm 1 indicated that it was about to produce the crispy cereal. Each Nash equilibrium is stable because once the strategies are chosen, no player will unilaterally deviate from them. However, without more information, we have no way of knowing which equilibrium (crispy/sweet vs. sweet/crispy) is likely to result—or if either will result. Of course, both firms have a strong incentive to reach one of the two Nash equilibria—if they both introduce the same type of cereal, they will both lose money. The fact that the two firms are not allowed to collude does not mean that they will not reach a Nash equilibrium. As an industry develops, understandings often evolve as firms “signal” each other about the paths the industry is to take.

Maximin Strategies

The concept of a Nash equilibrium relies heavily on individual rationality. Each player’s choice of strategy depends not only on its own rationality, but also on the rationality of its opponent. This can be a limitation, as the example in Table 13.4 shows.

In this game, two firms compete in selling file-encryption software. Because both firms use the same encryption standard, files encrypted by one firm’s software can be read by the other’s—an advantage for consumers. Nonetheless, Firm 1 has a much larger market share. (It entered the market earlier and its software has a better user interface.) Both firms are now considering an investment in a new encryption standard.

Note that investing is a dominant strategy for Firm 2 because by doing so it will do better regardless of what Firm 1 does. Thus Firm 1 should expect Firm 2 to invest. In this case, Firm 1 would also do better by investing (and earning $20 million) than by not investing (and losing $10 million). Clearly the outcome (invest, invest) is a Nash equilibrium for this game, and you can verify that it is the only Nash equilibrium. But note that Firm 1’s managers had better be sure that Firm 2’s managers understand the game and are rational. If Firm 2 should happen to make a mistake and fail to invest, it would be extremely costly to Firm 1. (Consumer confusion over incompatible standards would arise, and Firm 1, with its dominant market share, would lose $100 million.)

If you were Firm 1, what would you do? If you tend to be cautious—and if you are concerned that the managers of Firm 2 might not be fully informed or rational—you might choose to play “don’t invest.” In that case, the worst that can happen is that you will lose $10 million; you no longer have a chance of los- ing $100 million. This strategy is called a maximin strategy because it maximizes the minimum gain that can be earned. If both firms used maximin strategies, the outcome would be that Firm 1 does not invest and Firm 2 does. A maximin strategy is conservative, but it is not profit-maximizing. (Firm 1, for example, loses $10 million rather than earning $20 million.) Note that if Firm 1 knew for certain that Firm 2 was using a maximin strategy, it would prefer to invest (and earn $20 million) instead of following its own maximin strategy of not investing.

MAXIMIZING THE EXPECTED PAYOFF

If Firm 1 is unsure about what Firm 2 will do but can assign probabilities to each feasible action for Firm 2, it could instead use a strategy that maximizes its expected payoff. Suppose, for example, that Firm 1 thinks that there is only a 10-percent chance that Firm 2 will not invest. In that case, Firm 1’s expected payoff from investing is (.1)( 100) + (.9)(20) = $8 million. Its expected payoff if it doesn’t invest is (.1)(0) + (.9)( 10) = $9 million. In this case, Firm 1 should invest.

On the other hand, suppose Firm 1 thinks that the probability that Firm 2 will not invest is 30 percent. Then Firm 1’s expected payoff from investing is (.3)(100)+(.7)(20) = $16 million, while its expected payoff from not investing is (.3)(0) + (.7)(10) = $7 million. Thus Firm 1 will choose not to invest.

You can see that Firm 1’s strategy depends critically on its assessment of the probabilities of different actions by Firm 2. Determining these probabilities may seem like a tall order. However, firms often face uncertainty (over market conditions, future costs, and the behaviour of competitors), and must make the best decisions they can based on probability assessments and expected values.

4 Responses to “Game Theory”
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