6.1 Basic Concepts of Game Theory

The same factors that can lead to the formation of a monopoly — such as entry barriers or exclusive control over resources — can also give rise to a market structure in which few producers dominate supply: an oligopoly. Unlike perfect competition, in an oligopoly firms are large and have market power, just like a monopolist. However, unlike the monopolist, these firms compete with one another. Their decisions are therefore interdependent, since the choices of one firm directly affect the profits of the others. This interdependence makes the analysis of oligopoly different from that of the markets examined so far.

To understand the difference, let us return to the example introduced in the previous chapter: consider a market where the demand curve is given by $P=5-Q/1000$. Suppose that only two firms operate in the market — a duopoly — which we will call firm 1 and firm 2. Let $Q_1$ and $Q_2$ denote the quantities produced by the two firms, respectively. Assuming that both can freely choose their level of capital, as in the case of a monopolist, the marginal cost of each firm will be constant, say equal to 2.

To make optimal decisions, each firm must determine the quantity to produce that maximizes the difference between revenue and cost. Let us consider firm 1: its cost of producing any quantity $Q_1$ is simply $2Q_1$. To calculate revenue, we assume that the market price adjusts so that demand absorbs the overall quantity produced in the market. The price will then be $P=5-(Q_1+Q_2)/1000$, and thus firm 1’s revenue will be $[5-(Q_1+Q_2)/1000]\times Q_1$. At this point, a difficulty arises: how can firm 1 optimally choose $Q_1$ without knowing the quantity $Q_2$ produced by the other firm?

The tool for analyzing such situations is game theory. When the optimal decisions of an economic agent — in our case, a firm — depend on the choices of others, it becomes essential to anticipate the behavior of competitors. Game theory provides the tools to study these strategic situations, where outcomes depend on the interaction between the decisions of multiple rational agents.

A simultaneous-move game is a model that specifies three elements: who the players are, what strategies each player has at their disposal, and what the payoffs — in the case of firms, profits — are for each possible profile of strategies, that is, each combination of choices made by the players. It is assumed that players make their decisions independently and simultaneously, without knowing the other players’ moves in advance, and that each acts with the goal of maximizing their own payoff.

The following figure This game is an example of a prisoner’s dilemma. The idea comes from a hypothetical situation in which two suspects, interrogated separately, must decide whether to confess or remain silent. In our game, Down corresponds to confessing and Up to remaining silent for player 1; Right corresponds to confessing and Left to remaining silent for player 2. The combinations of choices determine the payoffs corresponding to the possible “sentences.” The abstract version of the game was formulated by Merrill Flood and Melvin Dresher at the RAND Corporation in 1949. Albert Tucker later introduced the prisoner story, making it famous in economics and psychology. provides an example of a game with two players and two strategies each. Player 1’s possible strategies are Up and Down, and player 2’s are Left and Right. There are thus four possible strategy profiles: (Up, Left), (Up, Right), (Down, Left), and (Down, Right). The corresponding payoffs are shown in blue for player 1 and in red for player 2.

Once the situation has been described through a game, the question we must answer is: what will the players do?

Dominated Strategies

A player’s strategy is dominated by another strategy of the same player if, regardless of the strategies chosen by the other players, the payoff obtained by playing the first strategy is lower than that obtained by playing the second. In other words, a dominated strategy is always worse than an alternative (always the same alternative), and therefore a rational player should never choose it.

In the game shown earlier, The prisoner’s dilemma reveals a fundamental tension between individual and collective interest. Each player has an incentive to follow their own dominant strategy, but if both do so the outcome is inefficient: the result is stable and rational from the individual point of view, but unfavorable for both compared to the cooperative strategy profile, namely (Up, Left). Up is dominated by Down, and Left is dominated by Right. Assuming that both players are rational and therefore do not choose dominated strategies, we can then predict that the outcome of the game will be the strategy profile (Down, Right). We have “solved” the game by eliminating the dominated strategies.

Pushing this logic further, we can assume that each player is not only rational, but also believes that the others are rational, that the others believe that they are rational, and so on. Proceeding in this way, we are often able to solve more complex games through a procedure of iterated elimination of dominated strategies. The procedure mirrors the logic described above: since players are rational, they should not choose dominated strategies. But not only that — players should also avoid strategies that, while not initially dominated, become dominated once the dominated strategies have been eliminated from the game. And so on. The following figure shows an example.

Iterated elimination - Step 1 Iterated elimination - Step 2 Reset

Returning to the duopoly we discussed at the beginning of the chapter, let us assume for simplicity that each firm has only four possible strategies: not to produce at all, to produce a medium-low quantity (400), a medium-high quantity (1200), or a high quantity (1600). Given that the market demand is $P = 5 - Q/1000$ and the marginal cost for each firm is equal to 2, the situation is described by the following game.

Although the game may seem complicated at first glance, with a bit of attention we can see that it can be solved by iteratively eliminating dominated strategies, following the procedure (in this case, in three steps) shown alongside.

Iterated elimination - Step 1 Iterated elimination - Step 2 Iterated elimination - Step 3 Reset

Nash Equilibrium

Not all games can be solved through the iterated elimination of dominated strategies. In many cases, multiple strategy profiles remain after the procedure, and it is not immediately clear what the outcome of the game will be. To address such situations, a more general idea is needed: the concept of Nash equilibrium. A strategy John Nash (Nobel Prize in Economics 1994) developed the concept of equilibrium in his doctoral thesis at Princeton in 1950. His advisor was Albert Tucker, the same who shortly before had made the prisoner’s dilemma famous. profile is a Nash equilibrium if no player has an incentive to unilaterally change their strategy, given the behavior of the others. In other words, each player is making the best choice for themselves, assuming the others do not change theirs. Nash equilibrium, a central concept in game theory, thus represents a situation in which players’ expectations and actions are mutually consistent.

The calculation of Nash equilibria is based on analyzing the players’ best responses. For each strategy the second player might choose, we identify the strategy (or strategies) that gives the first player the highest payoff. The same is then done for the other player. A Nash equilibrium is a strategy profile in which each strategy is a best response to the other: no player has an incentive to deviate if they believe the other won’t. When a game is represented in matrix form, as in the examples discussed earlier — two players with a finite number of strategies each — the calculation can be done easily by highlighting each player’s best responses in the cells of the matrix.

In the following figure, we illustrate a game that is not solvable through the iterated elimination of dominated strategies, but has a unique Nash equilibrium, namely (Up, Left).

Show player 1’s best reply

Show player 2’s best reply

As mentioned earlier, Nash equilibrium is a more general concept than the iterated elimination of dominated strategies. When a game can be solved through the iterated elimination of dominated strategies, the single outcome that survives the procedure is necessarily also the unique Nash equilibrium of the game. We illustrate this fact by revisiting the duopoly game presented earlier.

Show firm 1’s best reply Show firm 2’s best reply