Dominant and Dominated Strategies
Best Responses
Rationalizability and
Iterated Deletion of Dominated Strategies
One strategy strictly dominates another strategy
if it always yields a strictly higher payoff
no matter what the other players do.
A pure strategy is dominated for a player if
there’s some other strategy that that player could choose
which would give them a higher payoff
no matter what the other players are doing.
One strategy weakly dominates another strategy
if it never yields a strictly lower payoff
no matter what the other players do,
and sometimes yields a strictly higher payoff.
Which strategies are dominated?
Dominant Strategy Equilibrium
If each player has a single dominant strategy, then the dominant strategy equilibrium is the outcome of the game in which they each play their dominant strategy.
Dominant Strategy
If a player has a strategy which dominates all other strategies, it is called a dominant strategy.
Pick Up is dominated by Don’t:
no matter what the other person does,
you’re better off being a slob.
However, (PU, PU) Pareto dominates (D, D):
that is, both players would be better off
if they both picked up their stuff.
Two roommates independently choose whether to pick up own clothing from the floor.
Picking up your clothing has a $1 cost to you but generates a $2 benefit to your roommate.
The payoffs are specified in the matrix:
The First Strategic Tension:
Everyone doing what's best for themselves can lead to a group loss.
Dominated strategy: some other strategy is better
no matter what you believe the other players will do
Best response: the best strategy (or strategies) to play
given specific beliefs about what the other players will do
A strategy is rationalizable if it is a best response
for some belief about what the other player might do.
Note: a dominated strategy cannot be a best response, is not rationalizable
Informally: a thought exercise that says:
"If I believe the other player(s) are going to play some way, what should I do?"
Formally:
What is player 1's best response if they believe player 2 will play L?
What is player 1's best response if they believe player 2 will play R?
What is player 1's best response if they believe player 2 will play
L or R with equal probability?
What is player 1's best response if they believe player 2 will play
L with probability q,
and R with probability 1 - q?
How to search for a dominated strategy:
1. Look to see if it's dominated by another pure strategy
2. Look for candidate mixed strategies, especially different strategies with alternating large payoffs
3. Remember: you only need to find one strategy that dominates a strategy for strategy to be dominated.
Think about Brianna’s choice.
Does she have a dominated strategy?
What is the set of her rationalizable strategies?
Given that, we can eliminate Greek as a choice for Brianna…
Alex and Briana independently choose between a Greek and a Thai restaurant for lunch.
Briana prefers Thai to Greek and does not care whether she meets Alex or not.
Alex prefers Greek to Thai, but he has a crush on Briana and so mostly cares about running into her.
The payoffs are specified in the matrix:
We can remove from the game any strictly dominated strategies
(by either a pure or mixed strategy).
Removal may create new dominated strategies, remove them too…
The set of strategies that’s left at the end of that process
is the set of rationalizable strategies.
turns out to be independent of the order
in which dominated strategies are removed
Iterated Dominance
The process of eliminating strategies that are dominated, until no remaining strategies are dominated.
Rationalizable Strategies
The set of strategies that survive iterated dominance.
70% Game
Go to the link in your email to play.
Everyone guesses a number between 0 and 100. The closest guess to 70% of the class average will win 5 homework points!
Two hot dog carts on a beach; 9 possible locations.
Customers are evenly distributed along the beach and go to closest location.
What is the set of rationalizable strategies?
Everyone chooses an integer between 0 and 100.
The closest person to 70% of the average guess wins.
0
100
You are throwing a party.
Each person \(i\) contributes \(g_i\) anonymously.
From player \(i\)'s perspective:
Utility depends on \(G\) and private consumption \(x_i\):
Payoffs as functions of strategies:
Each player has income \(m_i\); private consumption is what they have after contributing.
Payoffs as functions of strategies:
Two person case: player 1 has \(m_1 = 32\), player 2 has \(m_2 = 40\).
What should player 1 contribute if she believes player 2 will contribute 10? 20?
50% probability of each?
Beliefs are a probability distribution
over all possible contribution levels by the other person.
So far we've mainly talked about what people won't do,
and we have a good predictor of what they will do --
but only if they have a single dominant strategy.
Next time: look for Nash Equilibrium
in which everyone is best responding to everyone else.