Mixed Strategy

Question 1

What is the mass population interpretation of mixed strategies?

Answer 1

Assuming heterogeneous population, p(ai) is the proportion of the population that will choose action ai.

Question 2

What is the general form of a Von-Neumann Morgenstern utility function?

Answer 2

U(a) = Σ p(a)u(a) for ALL actions a that are a member of the action set

Question 3

What is a lottery

Answer 3

Probability distribution over an outcome

Question 4

What is a deterministic outcome?

Answer 4

outcomes that occur with certainty

Question 5

What is a mixed strategy?

Answer 5

the probability distribution over a player’s actions

Question 6

What determines the player’s attitudes to risk?

Answer 6

The curvature of the utility functions. Expected utility fns do not remove this behaviour

Question 7

Why are V-NM preferences cardinal?

Answer 7

The size of the payoff now affects the size of expected payoffs,

Question 8

What is a definition of an MSNE?

Answer 8

A mixed strategy profile in a strategic game with V-NM preferences is an MSNE if for each player i and every possible mixed strategy of player i, the expected payoff to player i of the strategy is AT LEAST AS LARGE as the expected payoff to player i of all other mixed strategies, holding the mixed strategies of the other player constant. (best response definition)

The mixed strategy played by each player must be a best response to the others’ mixed strategies.

Question 9

Definition of strict domination (mixed strategy)

Answer 9

ai’’ strictly dominates ai’ iff
U(ai’‘,aj) > u(a’,aj)
for every list of actions aj that j can take

Question 10

Definition of weak dominance in a mixed strategy game?

Answer 10

ai’’ weakly dominates ai’ iff
U(ai’‘,aj) >= u(ai’, aj) for every possible aj
and
U(ai’‘,aj) > u(ai’,aj) for some certain values of aj

Question 11

State Brouwer’s Fixed Point Theorem.

Answer 11

Let X c R^m be a compact (closed and bounded) and convex set. Let f:X -> X be a continuous function. Then, there exists an x* that is a member of X s.t s* = f(x*)

Question 12

Brouwer’s Fixed Point Theorem in simple terms

Answer 12

• if the simplex is a closed and bounded set
• and a convex set
• continuous utility and best response functions on the simplex
• there will be a point which is mapped onto itself by the function. -> this is a fixed point that represents MSNE

Question 13

What might happen if the conditions of Brouwer’s fixed point theorem are not fulfilled?

Answer 13

Often, there will be no intersection of the best response functions e.g if there is discontinuity/ open/ non-convex sets

Question 14

What modification do we have to make to apply Brouwer to MSNE?

Answer 14

Brouwer requires that the function maps actions into the same space, but BR functions don’t do that

• we must modify the BR functions so that they map a vector of actions (ai,aj) from Ai x Aj into Ai x Aj space
• the fixed point that arises will be an MSNE

Question 15

How do the conclusions of Brouwer’s theorem change when we relax some of the assumptions?

Answer 15

1. Lack of continuous BR function doesn’t change the conclusions much, just use another Fixed Point Theorem
2. If there is an unbounded (infinite) set, cannot guarantee there will be MSNE existence in these cases
   The conditions we have outlined are SUFFICIENT, but not all NECESSARY