ECON 2259

Lowest to highest order of evolution and game theory

NOT NE > NE > ES [dominating strategy (can be weak or strict) > Strict NE

strict best response

A strategy that gives a player a strictly higher payoff than any other available option, given what the other players are doing.

Strict Nash Equilibrium:

An outcome where every player is playing a strict best response. If anyone changes their strategy, their payoff will definitively decrease.

evolutionary stability (ES)

the strategy s is called evolutionary stable if any small mutation playing a different strategy will die out

what was the lesson of the 2/3 average number game?

Understand your opponents’ payoff, sophistication, and their knowledge of your sophistication

common knowledge

if every player knows it, every player knows that every player knows it, every player knows that every player knows that every player knows it

how is mutual knowledge different from common knowledge

Every player knows a specific piece of information. However, players do not necessarily know what other players know. The chain of awareness stops at the very first layer

what does the median voter theorem prove

under specific conditions, the unique Nash equilibrium of this game occurs when both candidates choose the exact policy position of the median voter. This occurs because if either candidate deviates from the median position, the opponent can place their platform slightly closer to the center and capture more than half the electorate.

for any belief p that you hold about others’ choices…

you should choose the strategy that maximizes your payoff given that belief

why would i not choose the middle strategy in the penalty kick example

do not choose a strategy that is not a best response to any belief

how do you find the max (best response) of a partner’s payoff

take the derivative of a partner’s profit equation

what does it mean for an equilibria to be pareto ranked

A set of multiple Nash equilibria is Pareto ranked when they can be ordered by collective preference based on efficiency. In this situation, a Nash equilibrium yields a sub-optimal stable outcome for the players, whereas a Nash equilibrium exists that would provide higher payoffs to everyone if all players could collectively coordinate to switch strategies simultaneously.

what does it mean to have negative synergy in a partnership game and how does this effect the jointly optimal effort?

with negative synergy, each person’s effort reduces the marginal productivity of the other person’s effort, rather than literally making the other person’s total productivity negative. That’s why the jointly optimal effort is lower than it would be with positive synergy.

why can strategies that form a NE never be eliminated in iterative deletion of strictly dominated strategies?

Strategies that form part of a Nash equilibrium can never be eliminated by iterative deletion of strictly dominated strategies because, in a Nash equilibrium, each player’s strategy is a best response to the opponent’s strategy. A strictly dominated strategy can never be a best response to any opponent strategy. Therefore, a strategy that is part of a Nash equilibrium cannot be strictly dominated and cannot be eliminated through iterative deletion of strictly dominated strategies.

why can strategies that form a NE be weakly dominated

Strategies that form part of a Nash equilibrium CAN be weakly dominated because strategy (s) can still be a best response to some belief about Player 2’s strategy. For every strategy of Player 2, another strategy of Player 1 may give the same or a higher payoff than (s), and for at least one strategy of Player 2, it gives a strictly higher payoff.

how do i find the quantity produced under monopoly

The monopoly production function is equivalent to the marginal revenue by doubling the slope of the production function. Find where the monopoly production function (or MR) intersects with marginal cost.

what is the total demand function in bertrand competition

Q(pL) = a/b - pL/b

what is firm 1’s payoff in bertrand competition

u1 = p1q1 - cq1

what is firm 1’s payoff if p1 = p2 in bertrand competition

(p1-c)*[Q(pl)/2]

in bertrand competition if p2 = c, what is firm 1’s best response

if p2 =c, firm 1’s best response is to set any price the same or greater than p2 = c and also make 0 profit. any price below c will make negative profit.

what is the only nash equilibrium in betrand competition

the only nash equilibrium is for both firms to set p = mc

In the voter candidate mode, how far apart can two candidates be in equilibrium

Candidates have to be 1/6 < y < 5/6. Or else someone can stand up in the very middle and capture 1/3 of the vote, creating a 3 way tie of 1/3 of the vote.

what are lessons from the voter candidate model

• there are equilibria with more candidates as the benefit from winning increases or the cost of running decreases

payoff from a mixed strategy (pi)

pi is the weighted average of the expected payoffs of the pure strategies in the mix, where pi(si) is the weight on si

what has to be true for a player to use a mixed strategy

the payoffs between her choices are equal

what is the lesson of the auditing game (hint: changing payoffs in mixed NE)

when NE are mixed, changing payoffs only on one side, only changes equilibrium behavior on the other side

how do i know there are no other pure or mixed equilibria in a game matrix

I know there are no other pure or mixed Nash equilibria because iterative deletion of strictly dominated strategies leaves only one strategy for each player. Since a strictly dominated strategy cannot be played with positive probability in any Nash equilibrium, the only possible equilibrium is the remaining strategy profile.