Week 7 - Mixed Strategies

Question 1

what is a mixed strategy?

Answer 1

a probabilistic mix of a person’s strategies

Question 2

why do players in a zero sum game not want to use pure strategy?

Answer 2

because their opponent could predict and exploit that for their own gain

Question 3

what does the expected utility theory suggest about decision making?

Answer 3

that people make decisions based on expected utility rather than expected payoff

Question 4

two things to keep in mind when using expected utility theory?

Answer 4

• assume utility depends on the level of wealth (w) a person has u = u(w)
• individuals have a starting point from which they make their decision, their initial wealth w0

Question 5

initial wealth amount?

Answer 5

will be given; assume 0 if not given

Question 6

what is the expected utility?

Answer 6

the probability weighted average of the utilities of each lottery outcome

Question 7

expected utility formula?

Answer 7

U(L) = (P1 * U1) + (P2 * U2)

Question 8

how to calculate utility?

Answer 8

u(w) = put wealth into utility formula

Question 9

if the utility function is concave?

Answer 9

• individual is risk averse

- diminishing marginal utility of wealth

Question 10

if the utility function is linear?

Answer 10

• individual is risk neutral

- constant marginal utility of wealth

Question 11

if the utility function is convex?

Answer 11

• individual is risk-loving

- increasing marginal utility of wealth

Question 12

expected wealth formula?

Answer 12

(p)(payoff 1) + (1-p)(payoff 2)

essentially E(V) formula

Question 13

certainty equivalent definition?

Answer 13

the amount of money that makes someone indifferent between the lottery L and the CE

Question 14

certainty equivalent formula?

Answer 14

U(CE) = U(L)

sub CE into utility function and let it equal the expected utility of the lottery

Question 15

once you’ve found the answer for CE?

Answer 15

• if CE > U(L) = someone would have to pay you the difference to not play the lottery
• if CE < U(L) = you would pay the difference to avoid the lottery

Question 16

how would a risk neutral individual make decisions?

Answer 16

they would base decisions purely on expected values

Question 17

what to do when solving for a mixed strategy NE for each player?

Answer 17

we look at P1’s best response when P2 is mixing and plot on a graph: y-axis = P1’s E(P) and x-axis = P2’s q-mix

Question 18

how to write out someone’s best response rule?

Answer 18

• for q < POI
• for q = POI, all values of p are a best response for P2
• for q > POI

Question 19

how to find mutual BRR?

Answer 19

plot both on a small graph, looks like swastika and plot two straight lines for their POIs

Question 20

what is the NE in mixed strategies?

Answer 20

where the two players’ BRRs intersect

Question 21

beliefs a player should about have the other player’s strategy?

Answer 21

beliefs about what the other player’s strategy and probabilities is should be correct and chooses their best response according to these mixtures

Question 22

fundamental theorem of mixed strategy?

Answer 22

1) if a player is willing to mix between their pure strategies, they will be indifferent between them in equilibrium (same payoff)
2) any pure strategies used in equilibrium provide at least as high a payoff as all other pure strategies not used in the equilibrium mixed strategy

Question 23

how to use fundamental theorem?

Answer 23

• find person’s two payoff formulas, let them equal each other
• solve for p/q
• at the q we find, P1 will be indifferent between his strategies and will be willing to mix
• at the p we find, P2 will be indifferent between his strategies and will be willing to mix

Question 24

pure strategy analysis?

Answer 24

using best response method