Introduction to Game Theory: Basics, Dominance, Nash
Equilibrium
Economics 302 - Microeconomic Theory II: Strategic Behavior
Shih En Lu
Simon Fraser University
(with thanks to Anke Kessler)
ECON 302 (SFU)
Introduction to Game Theory
1 / 31
Topics
1
Game theory: basic de…nitions
2
(Strict) Dominance
3
Pure-strategy Nash equilibrium
4
Mixed-strategy Nash equilibrium
5
Best response graphs in 2x2 games
6
Mixed strategies and Dominance
ECON 302 (SFU)
Introduction to Game Theory
2 / 31
Most Important Things to Learn
1
De…nitions: strategy, strategy pro…le, dominant/dominated strategy,
iterated strict dominance (ISD), best response, Nash equilibrium (NE)
2
Determine the set of strategies that survive ISD
3
Find a game’s pure-strategy NE
4
Finding mixed-strategy NE in 2x2 games (or games that become so
after ISD)
5
Graphing best response functions in 2x2 games
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Introduction to Game Theory
3 / 31
Why Games?
We want to model strategic behaviour - i.e. how people act when
they realize that their own behaviour a¤ects others’choices and that
the outcome depends on others’actions as well as their own.
This …ts many economic situations where the "large number"
assumption of competitive markets fail: oligopoly, auctions,
bargaining, public goods, etc.
Also useful in other …elds: biology, political science, sports, etc.
ECON 302 (SFU)
Introduction to Game Theory
4 / 31
What is a Game?
A game has players that each chooses from actions available to
him/her.
Example: Bob can either go to class or skip class, and the professor
can either give a pop quiz or not.
This week, we study simultaneous-move games.
Simultaneous-move means that each player acts once, and does so
without knowing others’actions/strategies.
Example: The game would not be simultaneous-move if the professor
decides whether to give a quiz after seeing if Bob is in class.
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Introduction to Game Theory
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What is a Strategy (in a Simultaneous-Move Game)?
Each player plays a strategy, which is, in a simultaneous-move game,
a probability distribution over her actions.
Example: "Go to class" is one of Bob’s strategies. "Go to class with
probability 0.3 and Skip class with probability 0.7" is another.
A pure strategy is a strategy that puts probability 1 on a single action.
A mixed strategy is just any strategy. "Mixed" is used to emphasize
that the strategy may not be pure.
A collection of each player’s strategy is called a strategy pro…le.
Example: ((Go to class), (0.4 Give quiz, 0.6 No quiz)) is a strategy
pro…le.
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Introduction to Game Theory
6 / 31
Outcomes, Payo¤s and Complete Information
An outcome (or action pro…le) is a collection of actions taken by
each player.
Example: (Go to class, Give quiz) is an outcome.
Each outcome generates a utility, or payo¤, for each player. These
utilities are obtained from expected utility theory, so that a player
cares about her expected payo¤/utility.
Complete information means that each player knows every player’s
payo¤ from each outcome.
Example: If lazy students’payo¤s di¤er from hard-working ones’, and
the professor does not know whether Bob is lazy, then the game
would not feature complete information.
We will assume complete information roughly until the midterm.
ECON 302 (SFU)
Introduction to Game Theory
7 / 31
The Normal Form
A convenient way to represent a two-player simultaneous move game
of complete information is through the normal form (also known as
strategic form).
Prof.
Give Quiz No Quiz
Bob Go to class
0,0
2,6
Skip class
-5,-1
5,4
By convention, player 1 (Bob) picks the row, and player 2 (professor)
picks the column.
Each cell gives the payo¤s of player 1 and player 2, in that order.
What do you expect the professor to do? What about Bob?
ECON 302 (SFU)
Introduction to Game Theory
8 / 31
Dominance
For now, let’s only consider pure strategies.
A player’s strategy is (strictly) dominant if, for any combination of
actions by other players, it gives that player a strictly higher payo¤
than all her other strategies. ("The unique best choice no matter
what others do")
Examples: professor not giving a quiz.
A player’s strategy is (strictly) dominated if there exists another
strategy giving that player a strictly higher payo¤ for all combinations
of actions by other players. ("There’s something else that’s always
strictly better")
Examples: professor giving a quiz.
If a strategy is dominant, what can you say about that player’s other
strategies?
Is the converse true?
ECON 302 (SFU)
Introduction to Game Theory
9 / 31
Dominance Solvability (I)
It makes sense to predict that a player will play her dominant strategy
if she has one, and will never play a dominated strategy.
But in our example, while we can predict what the professor does,
Bob doesn’t have dominant or dominated strategies.
Idea: take it a step further, and assume Bob can also deduce what
the professor does.
Now we can predict what Bob will do.
ECON 302 (SFU)
Introduction to Game Theory
10 / 31
Dominance Solvability (II)
Iterated deletion of strictly dominated strategies, or iterated strict
dominance (ISD): after deleting dominated strategies, look at
whether other strategies became dominated with respect to the
remaining strategies. If so, delete these newly dominated strategies,
and repeat the process until no strategy is dominated.
Example: "Going to class" is initially not dominated, but after "Give
quiz" is eliminated, it becomes dominated.
A game is dominance solvable if ISD leads to a unique predicted
outcome, i.e. only one strategy for each player survives.
The Bob/Professor game is dominance solvable.
ECON 302 (SFU)
Introduction to Game Theory
11 / 31
Another Example
Cooperate Defect
Cooperate
-1,-1
-10,0
Defect
0,-10
-8,-8
This is an example of the famous Prisoner’s Dilemma.
"Cooperate" = keeping quiet; "Defect" = rat out the other prisoner
Is this game dominance solvable?
Is your predicted outcome Pareto e¢ cient?
Defecting gives both players a higher payo¤ no matter what the other
one does, but (Defect, Defect) is worse for both than (Cooperate,
Cooperate).
ECON 302 (SFU)
Introduction to Game Theory
12 / 31
The Prisoner’s Dilemma
In general, a PD is a 2x2 game where:
1
2
Both players have a dominant strategy.
The outcome where both players play their dominated pure strategy is
strictly better for both players than the outcome where they play their
dominant strategy.
Lesson: In some games, the outcome that you expect to occur
is ine¢ cient.
Can you think of other situations that can be modeled as a PD?
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Introduction to Game Theory
13 / 31
Exercise
Find the set of strategies that survive ISD.
Left Center Right
Top
1,7
1,1
7,0
Middle 5,3
6,4
5,1
Bottom 3,0
6,5
6,0
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Introduction to Game Theory
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Remarks on Dominance Solvability
A solution through ISD is not as convincing as a solution in dominant
strategies: it assumes that players correctly anticipate what others
will do.
However, ISD allows us to solve some games that don’t have a
dominant strategy for all players.
Moreover, the type of reasoning carried out in ISD seems feasible and
realistic, especially when the number of steps is small. So it’s still a
pretty appealing concept.
Fact: the order in which ISD is carried out (which can vary since
there can be more than one dominated strategy at a time) does not
in‡uence which strategies survive. (Brainteaser for math jocks: prove
this.)
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Introduction to Game Theory
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Many Games Are Not Dominance Solvable
It’s easy to come up with a game where ISD doesn’t help at all.
Colin
Pond Timmie’s
Rowena
Pond
1,1
0,0
Timmie’s
0,0
1,1
(This is a coordination game.)
Are we completely stuck?
ECON 302 (SFU)
Introduction to Game Theory
16 / 31
Nash Equilibrium (Motivation)
If Rowena and Colin haven’t set a meeting place, are meeting each
other for the …rst time and are new at SFU, then it’d be hard to
predict their behaviour.
But if they have communicated, have done this before or if there’s a
social norm, it seems likely that they will coordinate successfully.
Idea: We expect situations where everybody is playing her best
strategy, given others’strategies.
This way, nobody is making a mistake or has an incentive to switch.
Note: Just like for ISD, we are assuming that each player has correct
beliefs about what others are doing. But it’s a stronger assumption
here, because logic alone doesn’t allow us (or the players) to predict
behaviour.
ECON 302 (SFU)
Introduction to Game Theory
17 / 31
Nash Equilibrium (De…nition)
A player i’s strategy si is a best response to other players’strategies
if, taking as …xed these other players’strategies, si gives player i her
highest possible expected payo¤.
In other words, si is a best response if, given what others are doing,
no other strategy yields a strictly higher expected payo¤ for player i.
A Nash equilibrium (NE) is a strategy pro…le where every
player’s strategy is a best response to other players’strategies.
Again, this means that nobody has a reason to switch, giving what
everyone else is doing.
An NE in pure strategies or a pure-strategy NE is an NE where
every player’s strategy is pure.
Examples: (Skip Class, No Quiz) in …rst example; (Confess, Confess)
in PD; (Middle, Center) and (Bottom, Center) in exercise; (Pond,
Pond) and (Timmie’s, Timmie’s) in last example.
ECON 302 (SFU)
Introduction to Game Theory
18 / 31
Finding Pure-Strategy Nash Equilibria
Let’s …nd all the pure-strategy NEs in the following game:
Left Center Right
Palin
Top
0,1
0,0
5,2
-10,-10
Middle 1,1
4,3
4,0
-10,-10
Bottom 2,3
6,2
3,3
-10,-10
ECON 302 (SFU)
Introduction to Game Theory
19 / 31
Exercise
Find all the pure-strategy NEs in the following game:
Left Center Right
Top
3,7
9,6
1,6
Middle 0,0
4,0
2,0
Bottom 4,0
1,5
0,1
ECON 302 (SFU)
Introduction to Game Theory
20 / 31
Relating Dominance Solvability and Nash Equilibrium
Does a combination of strategies surviving ISD have to be a NE?
Can a player’s strategy in a NE involve an action eliminated by ISD?
Based on the above, which concept yields sharper predictions? Does
your answer depend on the game?
ECON 302 (SFU)
Introduction to Game Theory
21 / 31
Some Games Have No Pure-Strategy NE (I)
Rock Paper Scissors
Rock
0,0
-1,1
1,-1
Paper
1,-1
0,0
-1,1
Scissors -1,1
1,-1
0,0
But there might still be a mixed-strategy NE.
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Introduction to Game Theory
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Some Games Have No Pure-Strategy NE (II)
We saw that Rock-Paper-Scissors has no pure-strategy NE. Here’s
another example:
Kicker
Left
Right
Goalie Left
1,-1
-1,1
Right
-1,1
1,-1
(This game is sometimes called "matching pennies".)
What do soccer players actually do?
These are games where it’s important to keep the other(s) guessing.
ECON 302 (SFU)
Introduction to Game Theory
23 / 31
Optimality of Mixed Strategies
Kicker
Left
Goalie Left
1,-1
Right
-1,1
What would the goalie do
Right?
Right
-1,1
1,-1
if the kicker is more likely to play Left?
When will the goalie play both Left and Right with positive
probability?
In a NE, every action played with positive probability must be a
best response.
So if a player plays multiple actions with positive probability, he
must be indi¤erent between them, i.e. they must yield the
same expected utility.
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Introduction to Game Theory
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Finding Mixed-Strategy NE
Unfortunately, it is generally hard to …nd all mixed-strategy NE.
But in 2x2 games (2 players and 2 actions for each player), it’s not
too bad.
Kicker
Left
Right
Goalie Left
1,-1
-1,1
Right
-1,1
1,-1
Suppose the goalie plays Left with probability p. To …nd the NE, we
need to …nd the value of p that makes the kicker indi¤erent between
Left and Right.
Similarly, we need to …nd the mix of the kicker’s actions that makes
the goalie indi¤erent between Left and Right.
ECON 302 (SFU)
Introduction to Game Theory
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(Politically Incorrect) Example: Battle of the Sexes
Find all NE in the following game:
Girl
Ballet
Hockey
Guy
Ballet
3,1
0,0
Hockey
0,0
1,4
We can also …nd the expected payo¤s of each NE.
Note: for games with a …nite number of players and where each
player has a …nite number of actions, a NE always exists!
ECON 302 (SFU)
Introduction to Game Theory
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Bigger Games
Remember that NEs cannot involve strategies that are eliminated by
ISD (this is true of all NEs, not just pure-strategy ones).
So …nding all mixed-strategy NE is also feasible in games that reduce
down to 2x2 through ISD.
In games that don’t reduce that far, sometimes payo¤s are nice and
symmetric like in Rock-Paper-Scissors. But even then, it’s more work
than for 2x2 games.
Exercise (to do at home): Find all NE in RPS.
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Introduction to Game Theory
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Visualizing NE in 2x2 Games
We can graphically represent players’best responses (BR).
Guy
Ballet Hockey
Girl Ballet
3,1
0,0
Hockey
0,0
1,4
Let p be the probability of "Ballet" for the girl, and q be the
probability of "Ballet" for the guy.
What is the guy’s BR if p < 0.8? If p > 0.8? If p = 0.8?
What is the girl’s BR if q < 0.25? If q > 0.25? If q = 0.25?
The NE correspond to the intersection of the BR curves.
ECON 302 (SFU)
Introduction to Game Theory
28 / 31
Mixed Strategies and ISD (I)
Remember how we de…ned ISD: after deleting dominated strategies,
look at whether other strategies became dominated with respect to
the remaining strategies. If so, delete these newly dominated
strategies, and repeat the process until no strategy is dominated.
No mention that the strategies must be pure!
Here are some potential issues:
1
2
3
We’ve been deleting pure strategies. But there are mixed strategies
that place positive probability on those pure strategies. So can we
really ignore the strategies that have been deleted?
We’ve been deleting a strategy si if there’s another strategy si0 that
beats si for any pure strategy by the other player. But we’re only
supposed to delete si if we can …nd an si0 that beats si for any strategy,
pure or not, by the other player. Are we "overdeleting"?
We’ve been only deleting a strategy if it’s dominated by another pure
strategy. But we’re supposed to delete strategies that are dominated by
any strategy, pure or not. Are we "underdeleting"?
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Introduction to Game Theory
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Mixed Strategies and ISD (II)
1
2
3
If pure strategy si is dominated by pure strategy si0 , then any strategy
that uses si with positive probability is dominated by a strategy that
shifts that probability to si0 . Therefore, when we delete a pure
strategy si , we are also automatically deleting any mixed
strategy that places positive probability on that pure strategy
si , so we can really safely ignore si .
If si0 gives a higher payo¤ than si for any pure strategy by the other
player, then it also gives a higher expected payo¤ for any potentially
mixed strategy by the other player: a weighted average is higher if all
the numbers that you’re averaging are higher (and the probability
distribution - i.e. the other player’s mixed strategy - stays the same).
Therefore, si really is dominated, and we are not overdeleting.
Yes, we are underdeleting if we only consider pure strategies:
see problem set.
ECON 302 (SFU)
Introduction to Game Theory
30 / 31
Mixed Strategies and ISD (III)
Note: The observations on the previous slide all carry through with
more than two players, if instead of talking about "the other player’s
strategies," we talk about "the other players’strategy combinations"
(i.e. partial strategy pro…les that only include the others’strategies).
Note: We’re also "underdeleting" in another way: sometimes, a
mixed strategy can be dominated even if the pure strategies on which
it places positive probability are not dominated. But this is less
worrisome because it doesn’t make us underdelete pure strategies,
which means that it doesn’t "block" later steps of the ISD procedure.
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Introduction to Game Theory
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