Expectations & Randomization
Normal Form Games
Dominance
Normal Form Games
&
Dominance
Iterated Dominance
Expectations & Randomization
Normal Form Games
Dominance
Iterated Dominance
Let’s play the quarters game again
We each have a quarter. Let’s put them down on the desk at the
same time. If they show the same side (HH or TT), you take my
quarter. If they show opposite sides (HT or TH), I take yours.
Q: What is your expected payoff for playing H?
A: It depends on what you believe I will choose.
• Suppose that you think I will choose H with probability p.
• If you play H, you expect to win (50) with probability p and
lose with probability 1 − p
• So your expected value from H is
EV [H] = p · 50 + (1 − p) · 0 = 50p
Modeling choice under uncertainty: beliefs form, expected payoff
calculated, outcomes weighted by perceived likelihood
Expectations & Randomization
Normal Form Games
Dominance
Iterated Dominance
Let’s play the quarters game again
We each have a quarter. Let’s put them down on the desk at the
same time. If they show the same side (HH or TT), you take my
quarter. If they show opposite sides (HT or TH), I take yours.
But this time you have a third option (besides H and T): let your
neighbor flip the coin for you.
What do you choose? (H,T, or Flip)
Notice: Uncertainty can come from randomizing own strategy
(mixed strategy), not just strategic uncertainty (or external
randomness)
Expectations & Randomization
Normal Form Games
Dominance
Iterated Dominance
Part I: Games in Normal Form
Definition
A strategy is a complete contingent plan for a player in the game.
• Example: Pick the highest integer. What are the strategies of
each player?
• General notation: individual strategy space = Si , individual
strategy = si ∈ Si .
• A strategy profile: s ∈ S, where S = S1 × S2 × · · · × Sn
• Notation: i and −i. s = (si , s−i ).
Another element: Payoff function ui : S → R
Definition
A game in normal form (also called strategic form) consists of a
set of players, along with strategy spaces and payoff functions for
each player.
Expectations & Randomization
Normal Form Games
Dominance
Iterated Dominance
Matrix Games
Two-player, finite (as in S is finite) normal form games can be
described succinctly with bi-matrices. Take a look at your
handout. Let’s verify that the first matrix game satisfies the
requirements of the normal form.
Ok, before starting to analyze this game, let’s try playing one of
these. . .
Expectations & Randomization
Normal Form Games
Dominance
Iterated Dominance
Let’s Play
• Let’s play (g) for real money (see handout)
• For (g): odd perm number → choose A, B, or C; even perm
number → choose X, Y, or Z.
• Two separate clicker votes: first row players, then column
players.
• Now: I will select participants by drawing two cards for each
game, pay in $ according to strategies chosen
• Next time: I will select two more people and pay according to
clicker vote from today.
What kind of reasoning did you use?
Expectations & Randomization
Normal Form Games
Dominance
Iterated Dominance
How do clickers benefit us?
• Make it easy to play (some) in-class games
• Can help me see what you’ve learned
• Can help you see whether or not you understand
• Can help you see how far you’ve come
• Allow you to give me feedback/opinions
• Also: attendance/participation
Make sure your clicker is registered, bring it every class.
Expectations & Randomization
Normal Form Games
Dominance
Back to game a)
• How do you win at this game?
• Ask Dilbert
• What’s going on in this game?
Iterated Dominance
Expectations & Randomization
Normal Form Games
Dominance
Iterated Dominance
Concepts
Three concepts at play
• Beliefs: What Player i thinks all the other players will do
(θ−i ∈ ∆S−i ).
• Best response: A strategy is a best response if it is at least as
good as any other strategy, given beliefs.
• A strategy is dominated if there is another strategy that is
better for every set of beliefs. In other words, if Y is better
than X no matter what you think other players will choose,
then X is dominated by Y.
No rational person would ever choose a strategy that is dominated
Expectations & Randomization
Normal Form Games
Dominance
Iterated Dominance
Solution Concept: Dominance
• Assumes rationality: no one would ever choose an action this
is dominated by another.
• Apply to PD by crossing out all rows or columns that are
dominated
• Now let’s try to solve b)
• Does each player in game (b) have a dominant strategy?
• NO!
b) is not dominance solvable
Expectations & Randomization
Normal Form Games
Dominance
Iterated Dominance
What’s the problem?
• Need to further narrow down choices
• Rationality is too weak an a assumption
• Stronger: assume rationality is common knowledge
• Rational, know you are, know you know, know you know you
know, etc.
• Eliminate strategies through iterative process
Our second solution concept: Iterated Elimination of Dominated
Strategies
Expectations & Randomization
Normal Form Games
Dominance
Iterated Dominance
• Also known as Rationalizability
• Strategies that survived iterated elmination are called
rationalizable
• Try on b)
Now we get a unique outcome. Yay!
Iterated Dominance
Expectations & Randomization
Normal Form Games
Dominance
Iterated Dominance
Iterated Dominance: Weaknesses?
• Is iterated dominance THE ULTIMATE SOLUTION
CONCEPT?
• To see, let’s return to game c).
• Is c) dominance solvable?
• NO!
We can get more predictive power with stronger assumptions. Do
we want this?
Expectations & Randomization
Normal Form Games
Dominance
Iterated Dominance
Iterated Dominance: Are we missing anything?
Let’s try d)
• Seems like no dominance going on. Are we missing anything?
• Free your mind and dominance will follow
• If we enrich the strategy space to allow mixed strategies, then
B is dominated
• By what?
• (1/2, 0, 1/2)
• Now we continue iterated elimination
A strategy may be dominated by a combination of other strategies
Expectations & Randomization
Normal Form Games
Dominance
Iterated Dominance
Player 1’s expected payoff, as a function of beliefs about
Player 2’s strategy
Expectations & Randomization
Normal Form Games
Dominance
Iterated Dominance
Mixed strategies and best responses
Let’s look at e) now
• Is B dominated?
• No, not even by mixed strategies
• Is it ever a best response?
• Not to X or Y. . . ...
• But it is a BR to (1/2, 1/2) (one example)
• If a strategy is a best response to some beliefs, it can’t be
dominated
A strategy may be a best response (therefore undominated) even if
it is not the best response to any pure strategy