Dominance

A pure strategy si of player i is strictly dominated if there is a pure or mixed strategy, σi such that uii,s-i) > ui(si,s-i) for all s-i of the other players.

 

Discuss in the PD game.

 

Prisoners' dilemma version 2 Cartel breakdown

 

Company B

Price high

Price low

Company A

Price high

2,2

0,3

Price low

3,0

1,1

Pareto efficiency

A strategy profile s is PE if there does not exist another strategy profile s* such that ui(s*) ≥ ui(s) with for all i players and ui(s*) > ui(s) for at least one of the i players.

 

Discuss in the PD game and in the coordination game. Comment on Pareto characteristics of the different possible outcomes.

 

Coordination game

 

Company B

Adopt standard A

Adopt standard B

Company A

Adopt standard A

1,1

-1,-1

Adopt standard B

-1,-1

1,1

 

Pareto coordination game

 

Company B

Adopt standard A

Adopt standard B

Company A

Adopt standard A

2,2

-1,-1

Adopt standard B

-1,-1

1,1

 

 

Undominated strategies, UDi, for player i

The set of strategies in the strategy set of player i that is not dominated by another pure strategy or a mixed strategy of other non-dominated strategies.

Best response

A best response function requires a belief about what the other players are doing. For simplicity, we will discuss the issue in a two-player situation. Consider player 1 with a belief structure about player 2 of μ2. A strategy s1 is a best response for player 1 given μ2 if u1(s12) ≥ u1(s1*2) for all s1* distinct from s1 in S1.

 

Watson stresses the importance of the belief structure. Once it is specified the working out of a best response is a conventional economic calculation. In a sense developing a belief is akin to putting oneself into the other player’s shoes not only in knowing what options are available to him or her but how he or she might think about strategic alternatives.

Best response set over all beliefs of player i

Bi = {si| there exists a belief μ-i such that si is a best response for that belief}

 

Discuss problem 4 on p. 56 of Watson to illustrate.

Theorems

For 2 player finite games Bi and UDi are the same.

 

For n (>2) games, Bi is contained in the set UDi when beliefs are uncorrelated over the other n-1 players. With correlated beliefs the two sets are the same.

Common knowledge

Each player considers the other players to be rational. Each players knows the structure of the game (strategy spaces, payoffs, timing of moves, etc.) and knows that the other players know that and that the other players know that each has that knowledge etc. in an infinite chain.

Iterated removal of strictly dominated strategies, IRSDS

Each player reduces the domain of strategic interaction by removing all of the strictly dominated strategies of other players and of himself or herself and knows from common knowledge that the remaining domain of interaction will be the same for each player. In some instances the process reduces interactions so that there is only one strategy profile left, which is the IRSDS equilibrium of the game. If there is an IRSDS, it is also a Nash equilibrium and is unique.

A weakly dominated strategy

A pure strategy si of player i is weakly dominated if there is a pure or mixed strategy, σi such that uii,s-i) ≥ ui(si,s-i) for all s-i of the other players with the equality holding for some s-i but not all s-i.

Iterated removal of weakly dominated strategies, IRWDS

Each player reduces the domain of strategic interaction by removing all of the weakly dominated strategies of other players and of himself or herself and knows from common knowledge that the remaining domain of interaction will be the same for each player. In some instances the process reduces interactions so that there is only one strategy profile left, which is the IRWDS equilibrium of the game. This process may generate an equilibrium when no IRSDS equilibrium exists. Any IRWDS equilibrium is a Nash equilibrium but it may not be unique. Watson concentrates on the IRSDS process, but other authors deal with both. The IRWDS process, for example, is frequently used in auction theory to reveal a Nash equilibrium in a second-price auction.

Rationalizable strategies

The strategies that survive the IRSDS process are called rationalizable. If only one strategy profile is rationalizable it is the IRSDS equilibrium.

 

Note: we will discuss problems 4 and 6 in class.

Strategic uncertainty

The reduced game after the IRSDS process may still leave a situation in which players are not sure how the game will be played. Watson’s example is when the process of elimination leaves the players with a coordination game. The remaining game may also have two Nash equilibriums, one of which is inefficient compared to the other. His example of this case is the stag hunt (p. 62). We will talk about communication, sometimes referred to as “cheap talk,” and other factors that might prevent the choice of a strategy profile that generates an inefficient choice among equilibrium. These mechanisms do not help avoid inefficiency when there is a strategy profile that would generate a Pareto dominant outcome but this strategy profile is not a Nash equilibrium. This is the case, for example, with the PD game.

More on common knowledge

Example. Coloured hats (due to Geanakopolis)

Three girls are wearing hats that are either red or white. Each one can see the colour of the hats worn by the other but not the colour of her own hat. The hat of each is red.

 

An outsider tells them that there is at least one red hat. The first player cannot tell what colour her hat is, but her inability to answer, which we call a no, gives information to the other two. The no means that both of their hats cannot be white. The second individual also knows that if the hat worn by 3 were white she must have a red hat on as the possibilities given a no by 1 is that 2 and 3 are rw, rr or wr. Since she observes a r hat on 3 she does not know what the colour of her own hat is and answers a no. Her no provides additional information to the third person who knows that her hat cannot be white and therefore must be red.

 

3's ability to identify her own hat's colour does not give any information to the other two players as they already knew that the hat of the third person was red. If the questioning cycles, girls one and two can only give the same no answer as they cannot identify the colour of their own hats and the third still can answer yes. No more is learned by girls 1 and 2 after they hear two nos given in the first round.

 

Another way of expressing the reasoning: Individual 1 knows (wrr,rrr) while individual 2 knows (rwr, rrr) to begin with. After 1 answers “no”, 2 knows nothing more than before but if 3 had a white hat 2 would have known (rww, rrw) before 1 answered and (rrw) after and could have identified her hat as red. After 2's answer (given rrr is true state) 3 knows that 2's partition was (rwr, rrr) and therefore her hat is red. Before the answers of the other two her partition was (rrw, rrr). 2's inability to identify the colour of her hat means that rrw is not possible.

 

The common knowledge is the union of what each knows after the second girl answers “no”. 1 knows (rrr, wrr); 2 knows (rwr,rrr) and 3 knows rrr The common knowledge then is (wrr, rwr, rrr). After common knowledge is reached, 3 knows the colour of her hat and the other two girls do not.