## 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?

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.

## Input Output Analysis

#### Introduction

In its “static’ version, Professor Leontief’s input-output analysis deals with this particular question: “What level of output should each of the n industries in an economy produce, in order that it will just be sufficient to satisfy the total demand for that product?” The rationale for the term input-output analysis is quite plain to see. The output of any industry (say, the steel industry) is needed as an input in many other industries, or even for that industry itself; therefore the “correct’ (i.e., shortage-free as well as surplus-free) level of steel output will depend on the input requirements of all the n industries. In turn, the output of many other industries will enter into the steel industry as inputs, and consequently the “correct’ levels of the other products will in turn depend partly upon the input requirements of the steel industry. In view of this inter-industry dependence, any set of “correct” output levels for the n industries must be one that is consistent with all the input requirements in the economy. So,  input-output analysis should be of great use in production planning, such as in planning for the economic development of a country or for a program of national defense. Nevertheless, the problem posed in input-output analysis also boils down to one of solving a system of simultaneous equations, and matrix algebra can again be of service.

Assumptions

To simplify the problem, the following assumptions are as a rule adopted:

(1) each industry produces only one homogeneous commodity (broadly interpreted, this does permit the case of two or more jointly produced commodities, provided they are produced in a fixed proportion to one another).

(2) Each industry uses a fixed input ratio (or factor combination) for the production of its output.

(3) Production in every industry is subject to constant returns to scale, so that a k-fold change in every input will result in an exactly k-fold change in the output.

These assumptions are, of course, unrealistic. From these assumptions we see that, in order to produce each unit of the jth commodity, the input need for the ith commodity must be a fixed amount, which we shall denote by aij. Specifically, the production of each unit of the jth commodity will require a1j (amount) of the first commodity, a2j of the second commodity,…, and anj of the nth commodity. (The order of the subscripts in aij is easy to remember: the first subscript refers to the input, and the second to the output, so that aij, indicates how much of the ith commodity is used for the production of each unit of the jth commodity.)

For example, we may assume prices to be given and, thus, adopt “a dollar’s worth’ of each commodity as its unit. Then the statement a32= 0.35 will mean that 35 cents’ worth of the third commodity is required as an input for producing a dollar’s worth of the second commodity. The aij symbol will be referred to as an input coefficient.

For an n-industry economy, the input coefficients can be arranged into a matrix A = [aij], as in Table below, in which each column specifies the input requirements for the production of one unit of the output of a particular industry. The Second column, for example, states that to produce a unit (a dollar’s worth) of commodity II, the inputs needed are: a12 units of commodity I, a22 units of commodity II, etc. If no industry uses its own product as an input, then the elements in the principal diagonal of matrix A will all be zero.

To be Continued….

## The two-part tariff

The two-part tariff is related to price discrimination and provides another means of extracting consumer surplus. It requires consumers to pay a fee up front for the right to buy a product. Consumers then pay an additional fee for each unit of the product they wish to consume. The classic example of this strategy is an amusement park. You pay an admission fee to enter, and you also pay a certain amount for each ride. The owner of the park must decide whether to charge a high entrance fee and a low price for the rides or, alternatively, to admit people for free but charge high prices for the rides.

Example: The two-part tariff has been applied in many settings: tennis and golf clubs (you pay an annual membership fee plus a fee for each use of a court or round of golf); the rental of large mainframe computers (a flat monthly fee plus a fee for each unit of processing time consumed); telephone service (a monthly hook-up fee plus a fee for minutes of usage). The strategy also applies to the sale of products like safety razors (you pay for the razor, which lets you consume the blades that fit that brand of razor).

The problem for the firm is how to set the entry fee (which we denote by T) versus the usage fee (which we denote by P). Assuming that the firm has some market power, should it set a high entry fee and low usage fee, or vice versa? To solve this problem, we need to understand the basic principles involved.

##### SINGLE CONSUMER CASE

Let’s begin with the artificial but simple case illustrated in Figure below. Suppose there is only one consumer in the market (or many consumers with identical demand curves). Suppose also that the firm knows this consumer’s demand curve. Now, remember that the firm wants to capture as much consumer surplus as possible. In this case, the solution is straight forward: Set the usage fee P equal to marginal cost and the entry fee T equal to the total consumer surplus for each consumer. Thus, the consumer pays T* (or a bit less) to use the product, and P* = MC per unit consumed. With the fees set in this way, the firm captures all the consumer surplus as its profit.

##### TWO CONSUMERS CASE

Now suppose that there are two different consumers (or two groups of identical consumers). The firm, however, can set only one entry fee and one usage fee. It would thus no longer want to set the usage fee equal to marginal cost. If it did, it could make the entry fee no larger than the consumer surplus of the consumer with the smaller demand (or else it would lose that consumer), and this would not yield a maximum profit. Instead, the firm should set the usage fee above marginal cost and then set the entry fee equal to the remaining consumer surplus of the consumer with the smaller demand. Figure below illustrates this. With the optimal usage fee at P* greater than MC, the firm’s profit is 2T* + (P*−MC)(Q1+Q2). (There are two consumers, and each pays T*.) You can verify that this profit is more than twice the area of triangle ABC, the consumer surplus of the consumer with the smaller demand when P=MC. To determine the exact values of P* and T*, the firm would need to know (in addition to its marginal cost) the demand curves D1 and D2. It would then write down its profit as a function of P and T and choose the two prices that maximize this function.

##### MANY CONSUMERS CASE

Most firms, however, face a variety of consumers with different demands. Unfortunately, there is no simple formula to calculate the optimal two-part tariff in this case, and some trial-and-error experiments might be required. But there is always a trade-off: A lower entry fee means more entrants and thus more profit from sales of the item. On the other hand, as the entry fee becomes smaller and the number of entrants larger, the profit derived from the entry fee will fall. The problem, then, is to pick an entry fee that results in the optimum number of entrants—that is, the fee that allows for maximum profit. In principle, we can do this by starting with a price for sales of the item P, finding the optimum entry fee T, and then estimating the resulting profit. The price P is then changed, and the corresponding entry fee calculated, along with the new profit level. By iterating this way, we can approach the optimal two-part tariff.

Figure above illustrates this principle. The firm’s profit π is divided into two components, each of which is plotted as a function of the entry fee T, assuming a fixed sales price P. The first component, πa , is the profit from the entry fee and is equal to the revenue n(T)T, where n(T) is the number of entrants. (Note that a high T implies a small n.) Initially, as T is increased from zero, revenue n(T)T rises. Eventually, however, further increases in T will make n so small that n(T)T falls. The second component, πs , is the profit from sales of the item itself at price P and is equal to (P − MC)Q, where Q is the rate at which entrants purchase the item. The larger the number of entrants n, the larger Q will be. Thus πs falls when T is increased because a higher T reduces n.

Starting with a number for P, we determine the optimal (profit-maximizing) T*. We then change P, find a new T*, and determine whether profit is now higher or lower. This procedure is repeated until profit has been maximized.

Obviously, more data are needed to design an optimal two-part tariff than to choose a single price. Knowing marginal cost and the aggregate demand curve is not enough. It is impossible (in most cases) to determine the demand curve of every consumer, but one would at least like to know by how much individual demands differ from one another. If consumers’ demands for your product are fairly similar, you would want to charge a price P that is close to marginal cost and make the entry fee T large. This is the ideal situation from the firm’s point of view because most of the consumer surplus could then be captured. On the other hand, if consumers have different demands for your product, you would probably want to set P well above marginal cost and charge a lower entry fee T. In that case, however, the two-part tariff is a less effective means of capturing consumer surplus; setting a single price may do almost as well.

At Disneyland in California and Walt Disney World in Florida, the strategy is to charge a high entry fee and charge nothing for the rides. This policy makes sense because consumers have reasonably similar demands for Disney vacations. Most people visiting the parks plan daily budgets (including expenditures for food and beverages) that, for most consumers, do not differ very much.

Firms are perpetually searching for innovative pricing strategies, and a few have devised and introduced a two-part tariff with a “twist”—the entry fee T entitles the customer to a certain number of free units. For example, if you buy a Gillette razor, several blades are usually included in the package. The monthly lease fee for a mainframe computer usually includes some free usage before usage is charged. This twist lets the firm set a higher entry fee T without losing as many small customers. Because these small customers might pay little or nothing for usage under this scheme, the higher entry fee will capture their surplus without driving them out of the market, while also capturing more of the surplus of the large customers.

Peak-load pricing also involves charging different prices at different points in time. Rather than capturing consumer surplus, however, the objective is to increase economic efficiency by charging consumers prices that are close to marginal cost.

For some goods and services, demand peaks at particular times—for roads and tunnels during commuter rush hours, for electricity during late summer afternoons, and for ski resorts and amusement parks on weekends. Marginal cost is also high during these peak periods because of capacity constraints. Prices should thus be higher during peak periods.

This is illustrated in above Figure, where D1 is the demand curve for the peak period and D2 the demand curve for the nonpeak period. The firm sets marginal revenue equal to marginal cost for each period, obtaining the high price P1 for the peak period and the lower price P2 for the nonpeak period, selling corresponding quantities Q1 and Q2. This strategy increases the firm’s profit above what it would be if it charged one price for all periods. It is also more efficient: The sum of producer and consumer surplus is greater because prices are closer to marginal cost.

The efficiency gain from peak-load pricing is important. If the firms were a regulated monopolist (e.g., an electric utility), the regulatory agency should set the prices P1 and P2 at the points where the demand curves, D1 and D2, intersect the marginal cost curve, rather than where the marginal revenue curves intersect marginal cost. In that case, consumers realize the entire efficiency gain.

Note that peak-load pricing is different from third-degree price discrimination. With third-degree price discrimination, marginal revenue must be equal for each group of consumers and equal to marginal cost. Why? Because the costs of serving the different groups are not independent. For example, with unrestricted versus discounted air fares, increasing the number of seats sold at discounted fares affects the cost of selling unrestricted tickets—marginal cost rises rapidly as the airplane fills up. But this is not so with peak-load pricing (or for that matter, with most instances of intertemporal price discrimination). Selling more tickets for ski lifts or amusement parks on a weekday does not significantly raise the cost of selling tickets on the weekend. Similarly, selling more electricity during off-peak periods will not significantly increase the cost of selling electricity during peak periods. As a result, price and sales in each period can be determined independently by setting marginal cost equal to marginal revenue for each period.

Movie theaters, which charge more for evening shows than for matinees, are another example. For most movie theaters, the marginal cost of serving customers during the matinee is independent of marginal cost during the evening. The owner of a movie theater can determine the optimal prices for the evening and matinee shows independently, using estimates of demand and marginal cost in each period.

## Intertemporal price discrimination

Two other closely related forms of price discrimination are important and widely practiced. The first of these is inter-temporal price discrimination: separating consumers with different demand functions into different groups by charging different prices at different points in time. The second is peak-load pricing: charging higher prices during peak periods when capacity constraints cause marginal costs to be high. Both of these strategies involve charging different prices at different times, but the reasons for doing so are somewhat different in each case.

The objective of inter-temporal price discrimination is to divide consumers into high-demand and low-demand groups by charging a price that is high at first but falls later. To see how this strategy works, think about how an electronics company might price new, technologically advanced equipment, such as high-performance digital cameras or LCD television monitors.

In the above Figure, D1 is the (inelastic) demand curve for a small group of consumers who value the product highly and do not want to wait to buy it (e.g., photography buffs who want the latest camera). D2 is the demand curve for the broader group of consumers who are more willing to forgo the product if the price is too high. The strategy, then, is to offer the product initially at the high price P1, selling mostly to consumers on demand curve D1 .Later, after this first group of consumers has bought the product, the price is lowered to P2, and sales are made to the larger group of consumers on demand curve D2.

There are other examples of inter-temporal price discrimination. One involves charging a high price for a first-run movie and then lowering the price after the movie has been out a year. Another, practiced almost universally by publishers, is to charge a high price for the hardcover edition of a book and then to release the paperback version at a much lower price about a year later. Many people think that the lower price of the paperback is due to a much lower cost of production, but this is not true. Once a book has been edited and typeset, the marginal cost of printing an additional copy, whether hardcover or paperback, is quite low, perhaps a dollar or so. The paperback version is sold for much less not because it is much cheaper to print but because high-demand consumers have already purchased the hardbound edition. The remaining consumers—paperback buyers—generally have more elastic demands.