Chapter summary

In this chapter we present the theory of auctions, which is considered to be one of the most successful applications of game theory, and in particular of games with incomplete information. We mainly study symmetric auctions with independent private values and risk-neutral buyers. An auction is presented as a game with incomplete information and the main interest is in the (Bayesian) equilibrium of this game, that is, in the bidding strategies of the buyers and in the expected revenue of the seller. A hallmark of this theory is the Revenue Equivalence Theorem, which states that in any equilibrium of an auction method in which (a) the winner is the buyer with the highest valuation for the auctioned item, and (b) any buyer who assigns private value 0 to the auctioned item pays nothing, the expected revenue of the seller is independent of the auction method. This theorem implies that a wide range of auction methods yield the seller the same expected revenue. We also prove that the expected revenue to the seller increases if all buyers are risk averse, and it decreases if all buyers are risk seeking.

The theory is then extended to selling mechanisms. These are abstract mechanisms to sell items to buyers that include, e.g., post-auction bargaining between the seller and the buyers who placed the highest bids. We prove the revelation principle for selling mechanisms, which allows us to consider only a simple class of mechanisms, called incentive-compatible direct selling mechanisms.

Chapter summary

This chapter is devoted to the study of the nucleolus, which is, like the Shapley value, a single-point solution concept for coalitional games. The notion that underlies the nucleolus is that of excess: the excess of a coalition at a vector x in ℝN is the difference between the worth of the coalition and the total amount that the members of the coalition receive according to x. When the excess is positive, the members of the coalition are not content with the total amount that they together receive at x, which is less than the worth of the coalition. Each vector x in ℝN corresponds to a vector of 2N excesses of all coalitions. The nucleolus of a coalitional game relative to a set of vectors in ℝN consists of the vectors in that set whose vector of excesses are minimal in the lexicographic order. It is proved that the nucleolus relative to any compact set is nonempty and if the set is also convex, then the nucleolus relative to that set consists of a single vector.

The nucleolus of the game is the nucleolus relative to the set of imputations, that is, the set of efficient and individually rational vectors. The prenucleolus of a coalitional game is its nucleolus relative to the set of preimputations, that is, the set of all efficient vectors. Both the nucleolus and the prenucleolus are defined for any coalition structure.

Chapter summary

In this chapter we extend Aumann's model of incomplete information with beliefs in two ways. First, we do not assume that the set of states of the world is finite, and allow it to be any measurable set. Second, we do not assume that the players share a common prior, but rather that the players' beliefs at the interim stage are part of the data of the game. These extensions lead to the concept of a belief space. We also define the concept of a minimal belief subspace of a player, which represents the model that the player “constructs in his mind” when facing the situation with incomplete information. The notion of games with incomplete information is extended to this setup, along with the concept of Bayesian equilibrium. We finally discuss in detail the concept of consistent beliefs, which are beliefs derived from a common prior and thus lead to an Aumann or Harsanyi model of incomplete information.

Chapter summary

In this chapter we present a model of social choice, which studies how a group of individuals makes a collective choice from among a set of alternatives. The model assumes that each individual in the group holds a preference relation over a given set of alternatives, and the problem is how to aggregate these preferences to one preference relation that is supposed to represent the preference of the group. A function that maps each vector of preference relations to a single preference relation is called a social welfare function. The main result on this topic is Arrow's Impossibility Theorem, which states that every social welfare function that satisfies two properties, unanimity and independence of irrelevant alternatives, is dictatorial.

This impossibility result is then extended to social choice functions. A social choice function assigns to every vector of preference relations of all individuals in the group a single alternative, interpreted as the alternative that is most preferred by the group.

A social choice function is said to be nonmanipulable if no individual can manipulate the group's choice and obtain a better outcome by reporting a preference relation that is different from his true preference relation. Using the impossibility result for social choice functions we prove the Gibbard–Satterthwaite Theorem, which states that any nonmanipulable social choice function that satisfies the property of unanimity is dictatorial.

Chapter summary

This chapter introduces the concept of correlated equilibrium in strategic-form games. The motivation for this concept is that players' choices of pure strategies may be correlated due to the fact that they use the same random events in deciding which pure strategy to play. Consider an extended game that includes an observer who recommends to each player a pure strategy that he should play. The vector of recommended strategies is chosen by the observer according to a probability distribution over the set of pure strategy vectors, which is commonly known among the players. This probability distribution is called a correlated equilibrium if the strategy vector in which all players follow the observer's recommendations is a Nash equilibrium of the extended game.

The probability distribution over the set of strategy vectors induced by any Nash equilibrium is a correlated equilibrium. The set of correlated equilibria is a polytope that can be calculated as a solution of a set of linear equations.

In Chapters 4, 5, and 7 we considered strategic-form games and studied the concept of equilibrium. One of the underlying assumptions of those chapters was that the choices made by the players were independent. In practice, however, the choices of players may well depend on factors outside the game, and therefore these choices may be correlated. Players can even coordinate their actions among themselves.

A good example of such correlation is the invention of the traffic light: when a motorist arrives at an intersection, he needs to decide whether to cross it, or alternatively to give right of way to motorists approaching the intersection from different directions.

Chapter summary

This chapter presents the core, which is the most important set solution concept for coalitional games. The core consists of all coalitionally rational imputations: for every imputation in the core and every coalition, the total amount that the members of the coalition receive according to that imputation is at least the worth of the coalition.

The core of a coalitional game may be empty. A condition that characterizes coalitional games with a nonempty core is provided in Section 17.3. A game satisfying this condition is called a balanced game and the Bondareva–Shapley Theorem states that the core of a coalitional game is nonempty if and only if the game is balanced. This characterization is used in Section 17.4 to prove that every market game has a nonempty core. A game is called totally balanced if the cores of all its subgames are nonempty. It is proved that a game is totally balanced if and only if it is a market game. Similarly, a game is totally balanced if and only if it is the minimum of finitely many additive games.

In Section 17.6 it is proved that the core is a consistent solution concept with respect to the Davis–Maschler definition of a reduced game; that is, for every imputation in the core and every coalition, the restriction of the imputation to that coalition is in the core of the Davis–Maschler reduced game to that coalition.

Chapter summary

In this chapter we present the model of repeated games. A repeated game consists of a base game, which is a game in strategic form, that is repeated either finitely or infinitely many times. We present three variants of this model:

• The finitely repeated game, in which each player attempts to maximize his average payoff.

• The infinitely repeated game, in which each player attempts to maximize his long-run average payoff.

• The infinitely repeated game, in which each player attempts to maximize his discounted payoff.

For each of these models we prove a Folk Theorem, which states that under some technical conditions the set of equilibrium payoffs is (or approximates) the set of feasible and individually rational payoffs of the base game.

We then extend the Folk Theorems to uniform equilibria for discounted infinitely repeated games and to uniform ε-equilibria for finitely repeated games. The former is a strategy vector that is an equilibrium in the discounted game, for every discount factor sufficiently close to 1, and the latter is a strategy vector that is an ε-equilibrium in all sufficiently long finite games.

In the previous chapters, we dealt with one-stage games, which model situations where the interaction between the players takes place only once, and once completed, it has no effect on future interactions between the players. In many cases, interaction between players does not end after only one encounter; players often meet each other many times, either playing the same game over and over again, or playing different games.

Chapter summary

In this chapter we introduce a graphic way of describing a game, the description in extensive form, which depicts the rules of the game, the order in which the players make their moves, the information available to players when they are called to take an action, the termination rules, and the outcome at any terminal point. A game in extensive form is given by a game tree, which consists of a directed graph in which the set of vertices represents positions in the game, and a distinguished vertex, called the root, represents the starting position of the game. A vertex with no outgoing edges represents a terminal position in which play ends. To each terminal vertex corresponds an outcome that is realized when the play terminates at that vertex. Any nonterminal vertex represents either a chance move (e.g., a toss of a die or a shuffle of a deck of cards) or a move of one of the players. To any chance-move vertex corresponds a probability distribution over the edges emanating from that vertex, which correspond to the possible outcomes of the chance move.

To describe games with imperfect information, in which players do not necessarily know the full board position (like poker), we introduce the notion of information sets. An information set of a player is a set of decision vertices of the player that are indistinguishable by him given his information at that stage of the game.

Chapter summary

This chapter is devoted to a theory of repeated games with vector payoffs, known as the theory of approachability, developed by Blackwell in 1956. Blackwell considered two-player repeated games in which the outcome is an m-dimensional vector of attributes, and the goal of each player is to control the average vector of attributes. The goal can be either to approach a given target set S ⊆ ℝm, that is, to ensure that the distance between the vector of average attributes and the target set S converges to 0, or to exclude the target set S, that is, to ensure that the distance between the vector of average attributes and S remains bounded away from 0. If a player can approach the target set we say that the set is approachable by the player, whereas if the player can exclude the target set we say that it is excludable by that player. Clearly, a set cannot be both approachable by one player and excludable by the other player.

We provide a geometric condition that ensures that a set is approachable by a player, and show that any convex set is either approachable by one player or excludable by the other player.

Two applications of the theory of approachability are provided: it is used, respectively, to construct an optimal strategy for the uninformed player in two-player zero-sum repeated games with incomplete information on one side, and to construct a no-regret strategy in sequential decision problems with experts.

Chapter summary

This chapter presents the Shapley value, which is one of the two most important single-valued solution concepts for coalitional games. It assigns to every coalitional game an imputation, which represents the payoff that each player can expect to obtain from participating in the game. The Shapley value is defined by an axiomatic approach: it is the unique solution concept that satisfies the efficiency, symmetry, null player, and additivity properties. An explicit formula is provided for the Shapley value of a coalitional game, as a linear function of the worths of the various coalitions. A second characterization, due to Peyton Young, involves a marginality property that replaces the additivity and null player properties.

The Shapley value of a convex game turns out to be an element of the core of the game, which implies in particular that the core of a convex game is nonempty. Similar to the core, the Shapley value is consistent: it satisfies a reduced game property, with respect to the Hart–Mas-Colell definition of the reduced game.

When applied to simple games, the Shapley value is known as the Shapley–Shubik power index and it is widely used in political science as a measure of the power distribution in committees.

This chapter studies the Shapley value, a single-valued solution concept for coalitional games first introduced in Shapley [1953]. Shapley's original goal was to answer the question “How much would a player be willing to pay for participating in a game?”

Chapter summary

The most important solution concept in noncooperative game theory is the Nash equilibrium. When games possess many Nash equilibria, we sometimes want to know which equilibria are more reasonable than others. In this chapter we present and study some refinements of the concept of Nash equilibrium.

In Section 7.1 we study subgame perfect equilibrium, which is a solution concept for extensive-form games. The idea behind this refinement is to rule out noncredible threats, that is, “irrational” behavior off the equilibrium path whose goal is to deter deviations. In games with perfect information, a subgame perfect equilibrium always exists, and it can be found using the process of backward induction.

The second refinement, presented in Section 7.3, is the perfect equilibrium, which is based on the idea that players might make mistakes when choosing their strategies. In extensive-form games there are two types of perfect equilibria corresponding to the two types of mistakes that players may make: one, called strategic-form perfect equilibrium, assumes that players may make a mistake at the outset of the game, when they choose the pure strategy they will implement throughout the game. The other, called extensive-form perfect equilibrium, assumes that players may make mistakes in choosing an action in each information set. We show by examples that these two concepts are different and prove that every extensive-form game possesses perfect equilibria of both types, and that every extensive-form perfect equilibrium is a subgame perfect equilibrium.

Chapter summary

In this chapter we study situations in which players do not have complete information on the environment they face. Due to the interactive nature of the game, modeling such situations involves not only the knowledge and beliefs of the players, but also the whole hierarchy of knowledge of each player, that is, knowledge of the knowledge of the other players, knowledge of the knowledge of the other players of the knowledge of other players, and so on. When the players have beliefs (i.e. probability distributions) on the unknown parameters that define the game, we similarly run into the need to consider infinite hierarchies of beliefs. The challenge of the theory was to incorporate these infinite hierarchies of knowledge and beliefs in a workable model.

We start by presenting the Aumann model of incomplete information, which models the knowledge of the players regarding the payoff-relevant parameters in the situation that they face. We define the knowledge operator, the concept of common knowledge, and characterize the collection of events that are common knowledge among the players.

We then add to the model the notion of belief and prove Aumann's agreement theorem: it cannot be common knowledge among the players that they disagree about the probability of a certain event.

An equivalent model to the Aumann model of incomplete information is a Harsanyi game with incomplete information. After presenting the game, we define two notions of equilibrium: the Nash equilibrium corresponding to the ex ante stage, before players receive information on the game they face, and the Bayesian equilibrium corresponding to the interim stage, after the players have received information.

Chapter summary

In this chapter we construct the universal belief space, which is a belief space that contains all possible situations of incomplete information of a given set of players over a certain set of states of nature. The construction is carried out in a straightforward way. Starting from a given set of states of nature S and a set of players N we construct, step by step, the space of all possible hierarchies of beliefs of the players in N. The space of all possible hierarchies of beliefs of each player is proved to be a well-defined compact set T, called the universal type space. It is then proved that a type of a player is a joint probability distribution over the set S and the types of the other players. Finally, the universal belief space Ω is defined as the Cartesian product of S with n copies of T; that is, an element of Ω, called state of the world, consists of a state of nature and a list of types, one for each player.

Chapters 9 and 10 focused on models of incomplete information and their properties. A belief space Π with a set of players N on a set of states of nature S, is given by a set of states of the world Y, and, for each state of the world ωϵ Y, a corresponding state of nature s(ω) ϵ S and a belief πi (ω) ϵ Δ(Y) for each player i ϵ N.

Chapter summary

Given a game in strategic form we extend the strategy set of a player to the set of all probability distributions over his strategies. The elements of the new set are called mixed strategies, while the elements of the original strategy set are called pure strategies. Thus, a mixed strategy is a probability distribution over pure strategies. For a strategic-form game with finitely many pure strategies for each player we define the mixed extension of the game, which is a game in strategic form in which the set of strategies of each player is his set of mixed strategies, and his payoff function is the multilinear extension of his payoff function in the original game.

The main result of the chapter is the Nash Theorem, which is one of the milestones of game theory. It states that the mixed extension always has a Nash equilibrium; that is, a Nash equilibrium in mixed strategies exists in every strategic-form game in which all players have finitely many pure strategies. We prove the theorem and provide ways to compute equilibria in special classes of games, although the problem of computing Nash equilibrium in general games is computationally hard.

We generalize the Nash Theorem to mixed extensions in which the set of strategies of each player is not the whole set of mixed strategies, but rather a polytope subset of this set.