Appendix: Games with Imperfect Information

Figure 2.3:

Best-response functions for the mixed-action extended game

The proposition follows directly from the right-hand side of Figure 2.3 that shows that the two best-response functions given in (2.8) have a unique intersection.

Finally, the best-response functions given in (2.8) have a property of being composed of horizontal or vertical line segments. Since the equilibrium occurs when the two curves intersect in their

"middle" sections, we have it that under the NE mixed outcome, each player is indifferent to the choice among all other probabilities that can be played, assuming that the other player does not deviate from the mixed action. This result makes the intuitive interpretation of a mixed-action game rather difficult, because there is no particular reason why each player would stick to the mixed action played under the NE.

2.5 Appendix: Games with Imperfect Information

Games with imperfect information are brought up here only for the sake of completion, and the beginning readers are urged to skip this appendix. Games with imperfect information describe situations where some players do not always observe the action taken by another player earlier in the game, thereby making the player unsure which node has been reached. For example, Figure 2.4 describes a variant of the Pilot-Terrorist game given in Figure 2.1. In Figure 2.4 we suppose that the terrorist cannot monitor the direction in which the pilot is flying, say because the terrorist cannot read a compass or because the pilot disables some of the navigation equipment. The broken line connecting nodes II_C and II_N describes an information set for the terrorist. The information set tells us that in this game, the terrorist cannot distinguish whether node II_C or II_N has been reached. Thus, when the terrorist has her turn to make a move, she has to choose an action without knowing the precise node she is on. Formally,

Figure 2.4:

A game with imperfect information: Information sets

Definition 2.16

An information set for a player is a collection of nodes in which the player has to choose an action.

When a player reaches an information set, the player knows that the particular information set has been reached, but if the information set contains more than one node, the player does not know which particular node in this collection has been reached.

We now have the tools to define a game with imperfect information:

Definition 2.17

An extensive form game is called

1. A game with imperfect information if one of the information sets contains more than one node;

2. A game with perfect information if each information set contains a single node.

Thus, all the extensive form games analyzed in Section 2.2 are games with perfect information since each information set coincides with a single node.

We now slightly extend our definition of a strategy (Definition 2.8) to incorporate games with imperfect information:

Definition 2.18

In a game with imperfect information, a strategy for a player is a list of actions that a player chooses at any information set where the player is entitled to take an action.

Thus, Definition 2.18 provides a more general definition of a strategy (compared with Definition 2.8) since a strategy is a list of actions a player chooses in each information set rather than in each node where the player is entitled to take an action. Under perfect information, of course, Definitions 2.8 and 2.18 coincide, since under perfect information each information set is a singleton.

Finally, we need to extend our definition of subgames (Definition 2.9) to incorporate games with imperfect information.

Definition 2.19

A subgame is an information set that contains a single node, and all the subsequent decision and terminal nodes, provided that all subsequent nodes are not contained in information sets containing nodes that cannot be reached from the subgame.

Figure 2.5 illustrates a game with imperfect information. In Figure 2.5, the nodes labeled A, D, and G are starting nodes for a subgame. However, the nodes labeled B, C, E, and F are not starting nodes for a subgame since some subsequent nodes are contained in information sets containing nodes that cannot be reached from these nodes.

Figure 2.5:

Game with imperfect information: Subgames

For example, the modified Pilot-Terrorist game described in Figure 2.4 has only one subgame, which is the original game itself, because all subsequent nodes are contained in information sets containing more than one node.

We conclude our discussion of games with imperfect information with solving for NE and SPE for the modified Pilot-Terrorist game described in Figure 2.4. First, all the possible outcomes for this game are given by (NY, B), (NY, NB), (Cuba, B), and (Cuba, NB). Thus, in the Pilot-Terrorist game under imperfect information, the number of outcomes has been reduced from eight to four since the terrorist now takes a decision at one information set (compared with two nodes, under perfect information). Second, since this game does not have any proper subgames, any NE is also a SPE. Hence, in this case, the set of NE outcomes coincides with the SPE outcomes. Thus, we can easily conclude that (NY, NB) constitutes both NE and SPE outcomes.

2.6 Exercises

1. Using Definition 2.5,

(a) Write down the best-response functions for country 1 and country 2 for the Peace-War game described in Table 2.1, and decide which outcomes constitute NE.

(b) Write down the best-response functions for Jacob and Rachel for the game described in Table 2.3, and decide which outcomes constitute a NE (if there are any).

(c) Write down the best-response functions for player 1 and player 2 for the game described in Table 2.5, and decide which outcomes constitute a NE (if there are any).

2. Consider the normal form game described in Table 2.6. Find the conditions on the parameters a, b, c, d, e, f, g, and h that will ensure that

Normal form game: Fill in the conditions on payoffs

(a) the outcome (T, L) is a NE;

(b) the outcome (T, L) is an equilibrium in dominant actions;

(c) the outcome (T, L) Pareto dominates all other outcomes;

(d) the outcome (T, L) is Pareto noncomparable to the outcome (B, R).

3. Consider the Traveler's Dilemma, where two travelers returning home from a remote island where they bought identical rare antiques find out that the airline has managed to smash these rare

antiques. The airline manager assures the passengers of adequate compensation. Since the airline manager does not know the actual cost of the antiques, he offers the two travelers the opportunity to write down separately on a piece of paper the true cost of the antiques, which is restricted to be any number between 2 and 100.

Let n_i denote that value stated by traveler i, i = 1, 2, and assume that the travelers cannot communicate with each other during this game. The airline manager states the following

compensation rules: (a) If traveler i writes down a larger number than traveler j, (i.e., n_i > n_j), then he assumes that j is honest and i is lying. Hence, in this case, the airline manager will pay n_i - 2 to traveler i, and n_j + 2 to traveler j. Thus, the manager penalizes the traveler assumed to be lying and rewards the

one assumed to be honest. (b) If n_i = n_j, then the manager assumes that both travelers are honest and pays them the declared value of the antiques. Letting n₁ and n₂ be the actions of the players, answer the following questions:

(a) Under Definition 2.6, which outcomes are Pareto Optimal?

(b) Under Definition 2.4, which outcomes constitute a Nash equilibrium for this game.

4. Consider a normal form game between three major car producers, C, F, and G. Each producer can produce either large cars, or small cars but not both. That is, the action set of each producer i, i = C, F, G is . We denote by aⁱ the action chosen by player i, , and by πⁱ(a^C, a^F, a^G) the profit to firm i. Assume that the profit function of each player i is defined by

Answer the following questions.

(a) Does there exist a Nash equilibrium when ? Prove your answer!

(b) Does there exist a Nash equilibrium when ? Prove your answer!

5. Figure 2.6 describes an extensive form version of the Battle of the Sexes game given initially in Table 2.2. Work through the following problems.

(a) How many subgames are there in this game? Describe and plot all the subgames.

(b) Find all the Nash equilibria in each subgame. Prove your answer!

(c) Find all the subgame perfect equilibria for this game.

(d) Before Rachel makes her move, she hears Jacob shouting that he intends to go to the opera (i.e., play ω). Would such a statement change the subgame perfect equilibrium outcomes? Prove and explain!

6. (This problem refers to mixed actions games studied in the appendix, section 2.4.) Consider the Battle of the Sexes game described in Table 2.2.

(a) Denote by θ the probability that Jacob goes to the OPERA, and by ρ the probability that Rachel goes to the OPERA. Formulate the expected payoff of each player.

(b) Draw the best-response function for each player [R^J(ρ) and R^R(θ)].

(c) What is the NE in mixed actions for this game?

(d) Calculate the expected payoff to each player in this NE.

(e) How many times do the two best-response functions intersect? Explain the difference in the number of intersections between this game and the best-response functions illustrated in Figure 2.3.

Figure 2.6:

Battle of the Sexes in extensive form

2.7 References

Aumann, R. 1987. ''Game Theory.'' In The New Palgrave Dictionary of Economics, edited by J.

Eatwell, M. Milgate, and P. Newman. New York: The Stockton Press.

Axelrod, R. 1984. The Evolution of Cooperation. New York: Basic Books.

Binmore, K. 1992. Fun and Games. Lexington, Mass.: D.C. Heath.

Friedman, J. 1986. Game Theory with Applications to Economics. New York: Oxford University Press.

Fudenberg, D., and J. Tirole. 1991. Game Theory. Cambridge Mass.: MIT Press.

Gibbons, R. 1992. Game Theory for Applied Economists. Princeton, N.J.: Princeton University Press.

McMillan, J. 1992. Games, Strategies, and Managers. New York: Oxford University Press.

Moulin, H. 1982. Game Theory for the Social Sciences. New York: New York University Press.

Osborne, M., and A. Rubinstein. 1994. A Course in Game Theory. Cambridge, Mass.: MIT Press.

Rasrnusen, E. 1989. Games and Information: An Introduction to Game Theory. Oxford: Blackwell.

Chapter 3

Technology, Production Cost, and Demand

Large increases in cost with questionable increase in performance can be tolerated only for race horses and fancy [spouses].

—Lord Kelvin 1824-1907 (President of the Royal Society)

This chapter reviews basic concepts of microeconomic theory. Section 3.1 (Technology and Cost) introduces the single-product production function and the cost function. Section 3.2 analyzes the basic properties of demand functions. The reader who is familiar with these concepts and properties can skip this chapter and proceed with the study of industrial organization. The student reader should note that this chapter reflects the maximum degree of technicality needed to grasp the

material in this book. Thus, if the reader finds the material in this chapter to be comprehensible, then the student should feel technically well prepared for this course.