Games in normal form

The standard example to illustrate the normal form representation of a simple game is the prisoner’s dilemma, which is arguably the prototypical example of a two-player game. The prisoner’s dilemma has been phrased in many different ways;¹² in the next example we use what is closest to a business management setting.

Example 14.4 (Prisoner’s dilemma) Consider two firms, say, A and B, which have to set the prices at which their products are sold. The products are equivalent, so price is a major determinant of sales. On the one hand, firm A would like to keep its price high, to keep revenues high as well; on the other one, if the competitor firm B lowers its price, it will erode the market share of firm A. Indeed, sometimes a price war erupts, reducing profits for both firms. To represent the problem, let us assume that there are only two possible prices, low and high. Hence, we consider only a discrete set of possible actions; it is also possible to formalize a game with a continuum of actions represented by real numbers. The following outcomes could result, depending on firms’ actions:

• If firm A sets a high price and firm B sets a low price, firm A will be wiped off the market and firm B will get a huge reward; we obtain a symmetric outcome if we swap firms’ actions.
• If both firms set a low price, the result of this price war will be a fairly low profit for both firms.Table 14.1 Representation of prisoner’s dilemma in normal form.
• If firms collude and both of them set a high price, the result will be a fairly high profit for both of them.

Depending on the action selected by all of the players, they will receive a payoff. Unlike the optimization models that we have described the payoff for each player is a function of the decisions of all of the players. Representing the game in normal form requires to specify the payoff to each player, for any combination of actions. The normal form of prisoner’s dilemma is illustrated in Table 14.1. Firm A is the row player and firm B is the column player. Each cell in the table shows the payoff to firms A and B, respectively; the first number is the payoff for the row player, and the second number is the payoff to the column player:

• If both firms play high, the payoff is 2 for both of them.
• If firm A plays high, but firm B plays low, the payoff is 0 for the former and 3 for the latter; these payoffs are swapped if firm A plays low, but firm B plays high.
• If both firms play low, the ensuing price war results in a payoff 1 for each firm.

Actions are selected simultaneously, and we need a sensible way of predicting the result of the game.

It is easy to see that the normal form of a game is appropriate for a small and nonsequential game, and different representations might be more suitable in other cases. In particular, sequential games involve the selection of multiple actions by each player; each action, in general, may depend on the previous choices by the other players. When there is a discrete set of actions available at each stage of the game, this can be represented in extensive form by a tree, much like the decision trees of Section 13.1. Indeed, decision trees may be regarded as a multistage game between the decision maker and nature, which randomly selects an outcome at each chance node. Solving a decision tree requires the specification of a strategy, i.e., a selection of a choice for each decision node. By a similar token, a multistage game requires the specification of a strategy for each player, i.e., a mapping from each state/node in the tree to the set of available actions of the player that must make a choice at that stage of the game. We will not consider the extensive form of a game in this book. Furthermore, we consider only pure strategies, whereby one action is selected by a player. In mixed strategies, each action is selected with a certain probability. Even though we neglect these more advanced concepts, we see that actions are only the building blocks of strategies. Therefore, in the following text we will use the latter term, even though in our very simple examples actions and strategies coincide.

Now we are able to formalize a simple game involving n players; to specify such a game we need:

• The set of available strategies S_i for each player, i = 1, …, n; in other words, we need the set of available strategies for each player.
• The set of payoff functions π_j(s₁, s₂, …, s_n) for each player j = 1, …, n; the payoff depends on the set of strategies s_i ∈ S_i selected by all of the players, within the respective feasible set S_i.

Note that, given a set of strategies, the payoff is known, as there is no uncertainty involved; furthermore, each player knows the set of available strategies of the other players, as well as their payoff functions. Hence, all of the players have a clear picture of the incentives of the other players and there is no hidden agenda. Clearly, this is only the simplest kind of game one can consider; partial information and uncertainty are involved in more realistic models. Now the problem is to figure out which outcome should be expected as a result of the game, where by outcome we mean a vector of strategies , one per player.