Static games with complete information III
Static games with complete information III
ECOS3012 LECTURE 3
August 24, 2021
Recap
Firm competitions:
Cournot: in quantities
Bertrand: in prices
Hotelling: in attributes
Tragedy of the commons
This week
Expected utility
Mixed-strategy Nash equilibrium
Existence theorem of Nash equilibrium
Motivating example: penalty kick
From week 1’s lecture:
No equilibrium
Interpretation: there is no (deterministic) strategy profile where no one wants to deviate
What’s your prediction?
MIX: make actions unpredictable
Goalie
Left Right
Kicker Left 0, 1 1, 0
Right 1, 0 0, 1
Go back to the definitions
Set of players: unchanged
Strategy set for each player:
Include all probability distributions
A mixed strategy such that
for all .
.
How about payoff functions?
for each and combinations of ’s.
Questions: when are those numbers (from week 1) enough?
Expected utility
Question: when can we use a just few numbers (for payoffs of pure strategy outcomes) to represent a decision maker’s preference over uncertain outcomes?
Such that if (Left, Left), (Left, right), (Right, Left), (Right, Right) happen with probability and
Goalie
Left Right
Kicker Left a, b c, d
Right e, f g, h
Expected utility
possible events (outcomes)
An uncertain outcome is called a lottery,
: the player weakly prefers over
Answer: we can compare lotteries with expected utilities if four conditions are satisfied.
Completeness
Transitivity
Continuity
Independence
Expected utility
Completeness: the player can always rank any two lotteries
or or both
Transitivity
and implies that
Continuity
: changes a little bit, preference for changes a little bit
Independence:
If , then for any ,
vNM expected utility theorm
There exists utility (number) for each event such that
if and only if the players preference over all the lotteries satisfy completeness, transitivity, continuity and independence.
We can describe a game with only the pure-strategy outcome utilities
Goalie
Left Right
Kicker Left a, b c, d
Right e, f g, h
Definition of a game
Set of players: unchanged
Strategy set for each player:
A mixed strategy such that
for all .
.
Payoff functions:
Specify payoffs for pure strategy profiles:
Utility from mixed strategy profiles is the expected utility
Mixed-strategy NE
In the two-player normal-form game , the mixed strategy profile is a Nash equilibrium if each player’s mixed strategy is a best response to the other player’s mixed strategy.
Example: penalty kick
Find the mixed strategy NE
Key: if a player mixes between two actions, she must be indifferent
Goalie
q 1-q
Left Right
Kicker p Left 0, 1 1, 0
1-p Right 1, 0 0, 1
Example: battle of the sexes
Pure strategy NE:
(Opera, Opera) and (Fight, Fight)
Mixed NE?
Bob
q 1-q
Opera Fight
Ann p Opera 2, 1 0, 0
1-p Fight 0, 0 1, 2
Example: Stag hunt
Mixed NE: p* = ?, q* = ?
Hunter 2
q 1-q
Stag Hare
Hunter 1 p Stag 100, 100 0, 0
1-p Hare 99, 0 99, 99
Now that we have mixed strategies…
Revisit dominated strategies
Steps to find all Nash equilibria
Existence of Nash equilibrium
Revisit: dominated strategies
A pure strategy can be strictly dominated by a mixed strategy, even if it is not strictly dominated by any pure strategy.
B is not SDed by either T or M
However, it is strictly dominated by the mixed strategy (0.5, 0.5, 0)
Player 2
L R
T 3, – 0, –
Player 1 M 0, – 3, –
B 1, – 1, –
Revisit: dominated strategies
A pure strategy can be a best response to a mixed strategy, even if it is not a best response to any pure strategy.
B is never a best response when player 2 plays a pure strategy.
B is the best response when player 2 plays the mixed strategy (½ L, ½ R)
Player 2
L R
T 3, – 0, –
Player 1 M 0, – 3, –
B 2, – 2, –
Weak vs strict dominance
Two pirates, each holding half of a key to open passage to a hidden island.
Each chooses whether to travel to the hidden island or stay at home
“Home” is weakly dominated by “Island”
(Home, Home) is a NE
Lesson: Do not delete weakly dominated strategies if trying to find NE.
Pirate 2
Island Home
Pirate 1 Island 1, 1 0, 0
Home 0, 0 0, 0
Steps to find all Nash equilibria
Simplifying the game by deleting all strictly dominated strategies. If this leads to a unique strategy profile, then it is the unique Nash equilibrium of the game.
Find all pure strategy NE of the simplified game.
Assign probabilities to each player’s strategies in the simplified game and solve the mixed-strategy NE (of there is any).
Example: find all NE
IESDS:
Pure-strategy NE:
Mixed-strategy NE: use (p, q) graph
Player 2
L C R
T 5, 7 2, 8 0, 3
Player 1 M 0, 4 1, 6 2, 5
B 9, 2 2, 2 4, 1
Existence of Nash equilibrium
Nash 1950
In the n-player normal form game if
n is finite, and
is finite for each I,
then there exists at least one Nash equilibrium, possibly involving mixed strategies.
Proof not required.
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