The Value of a Coordination Game

The value of a game is the payoff a player can expect (ex ante) from playing the game. Understanding how the value changes with economic primitives is critical for policy design and welfare. However, for games with multiple equilibria, the value is difficult to determine. We therefore develop a new theory of the value of coordination games. The theory delivers testable comparative statics on the value and delivers novel insights relevant to policy design. For example, policies that shift behavior in the desired direction can make everyone worse off, and policies that increase everyone's payoffs can reduce welfare.

“Stunts” and Ski Games1

This chapter discusses ski games. It explains that games on skis have great impact. In addition to benefiting mental health, they are an important factor in physical training and the development of dexterity, strength, and the well-being of participants. The educational significance is well-known to all, whether a person plays against others or plays within a team. Furthermore, many good qualities are fostered through participation in games: attention, presence of mind, self-confidence, fairmindedness, willingness to help one another, self-denial, and composure. On the physical side, games can show how proficiently one manage one's skis; physiologically, blood circulation accelerates, breathing intensifies, the nervous system gains strength, and a person grows strong and hardy. There is also the value of a game as entertainment, recreation, and as a means of mood enhancement. With these benefits in mind, the chapter suggests a few ski stunts and games.

A TWO-STEP SHAPLEY VALUE FOR COOPERATIVE GAMES WITH COALITION STRUCTURES

In this paper, we study cooperative games with coalition structures. We show that a solution concept that applies the Shapley value to games among and within coalitions and in which the pure surplus that the coalition obtains is allocated among the intra-coalition members in an egalitarian way, is axiomatized by modified axioms on null players and symmetric players and the usual three axioms: efficiency, additivity and coalitional symmetry. In addition to the original definition, we give two expressions of this solution concept. One is an average of modified marginal contributions and the other is the weighted Shapley value of a game with restricted communication.

An optimal sequential procedure for a buying-selling problem with independent observations

We consider a buying-selling problem when two stops of a sequence of independent random variables are required. An optimal stopping rule and the value of a game are obtained.

An optimal sequential procedure for a buying-selling problem with independent observations

We consider a buying-selling problem when two stops of a sequence of independent random variables are required. An optimal stopping rule and the value of a game are obtained.

Repeated matrix games

A matrix game is played repeatedly, with the actions taken at each stage determining both a reward paid to Player I and the probability of continuing to the next stage. An infinite history of play determines a sequence (Rn ) of such rewards, to which we assign the payoff lim supn (R 1 + · ·· + Rn ). Using the concept of playable strategies, we slightly generalize the definition of the value of a game. Then we find sufficient conditions for the existence of a value and for the existence of stationary optimal strategies (Theorems 8 and 9). An example shows that the game need not have a value (Example 4).

Repeated matrix games

A matrix game is played repeatedly, with the actions taken at each stage determining both a reward paid to Player I and the probability of continuing to the next stage. An infinite history of play determines a sequence (Rn) of such rewards, to which we assign the payoff lim supn (R1 + · ·· + Rn). Using the concept of playable strategies, we slightly generalize the definition of the value of a game. Then we find sufficient conditions for the existence of a value and for the existence of stationary optimal strategies (Theorems 8 and 9). An example shows that the game need not have a value (Example 4).