Modeling Strategic Behavior

A Graduate Introduction to Game Theory and Mechanism Design

George J Mailath

• Publisher: WSPC
• Year: 2018
• ISBN:
  9789813239951
• Topic: Economía
• Subtopic: Microeconomía

This book provides a graduate level introduction to the economic modeling of strategic behavior. The goal is to teach economics doctoral students the tools of game theory and mechanism design that all economists should know. These tools play a critical role in economic research.

Contents:

• Normal and Extensive Form Games
• A First Look at Equilibrium
• Games with Nature
• Nash Equilibrium: Existence and Foundations
• Nash Equilibrium Refinements in Dynamic Games
• Signaling
• Repeated Games
• Topics in Dynamic Games
• Bargaining
• Introduction to Mechanism Design
• Dominant Strategy Mechanism Design
• Bayesian Mechanism Design
• Principal Agency
• Appendices

Readership: Graduate and doctoral students interested in microeconomics, game theory, and mechanism design. Game Theory;Microeconomics;Mathematical Economics;Econometrics;Mechanism Design;Nash Equilibrium;Dynamic Games0 Key Features:

• This book is based on the lecture notes of the first-year graduate course on game theory and mechanism design taught at the Economics Department, University of Pennsylvania, for many years by George J Mailath

Chapter 1

Normal and Extensive Form Games

1.1Normal Form Games

Most introductions to game theory start with the prisoner’s dilemma.¹ Two suspects (I and II) are separately interrogated. The prosecutors have sufficient evidence to convict each of a minor offence, but wish to convict them of a major offence. The potential results of the interrogation are illustrated in Figure 1.1.1. Clearly, no matter what the other suspect does, it is always better to confess than not confess.

This game is often interpreted as a partnership game, in which two partners simultaneously choose between exerting effort and shirking. Effort E produces an output of 6 at a cost of 4, while shirking S yields no output at no cost. Total output is shared equally. The result is given in Figure 1.1.2. With this formulation, no matter what the other partner does (E or S), the partner maximizes his/her payoff by shirking.

The scenarios illustrated in Figures 1.1.1 and 1.1.2 are examples of normal form games.

Definition 1.1.1. An n-player normal (or strategic) form game G is an n-tuple {(S₁, U₁), . . . , (S_n, U_n)}, where for each player i,

•S_i is a nonempty set, i’s strategy space, with typical element s_i, and

•

, i’s payoff function.

Figure 1.1.1:The prisoner’s dilemma, with the numbers describing length of sentence (the minus signs indicate that longer sentences are less desirable). In each cell, the first number is player I’s sentence, while the second is player II’s.

Figure 1.1.2:The prisoner’s dilemma, as a partnership game. In each cell, the first number is the row player’s payoff, while the second number is the column player’s payoff.

The normal form game G is finite if n < ∞ and |S_i| < ∞ for all i.

Notation: The set of strategy profiles is

, with a strategy profile denoted by s ≔ (s₁, . . . , s_n) ∈ S. The strategy profile omitting player i’s strategy is s_−i ≔ (s₁, . . . , s_i−1, s_i+1, . . . , s_n) ∈ S_−i ≔ ∏_k≠i S_k. Finally,

, so that s = (s_i, s_−i).

We sometimes write payoffs as a vector-valued function U : S →

ⁿ, or when S is finite, as the vector U ∈

^n|S| (recall that a vector x ∈

^k can be viewed as the function x : {1, . . . , k} →

, and conversely, a function from the finite set {1, . . . , k} to a set Y can be viewed as a vector in Y^k).

Example 1.1.1 (Sealed-bid second-price auction). Two bidders simultaneously submit bids (in sealed envelopes) for an object. A bid is a nonnegative number, with i’s bid denoted by b_i ∈

₊. Bidder i’s value for the object (reservation price, willingess to pay) is denoted by υ_i. The object is awarded to the highest bidder, who pays the second highest bid. Ties are resolved by a fair coin toss. Then, n = 2, S_i =

₊, and (taking expectations)

Example 1.1.2 (Sealed bi...