Introduction to Game Theory

Simultaneous Games

Dante Yasui

2024

Outline

• Game Tables/Strategic Form

• Nash Equilibrium

• Dominant Strategies

Split or Steal

Pause at 2:20. We’ll come back to Nick and Ibrahim!

Split or Steal

From the perspective of Nick, his payoffs depend on what Ibrahim has chosen:

	if Ibrahim Splits	if Ibrahim Steals
if Nick Splits he wins:	\(\$6,800\)	\(\$ 0\)
if Nick Steals he wins:	\(\$13,600\)	\(\$ 0\)

Split or Steal

From the perspective of Ibrahim, his payoffs depend on what Nick has chosen:

	if Nick Splits	if Nick Steals
if Ibrahim Splits he wins:	\(\$6,800\)	\(\$ 0\)
if Ibrahim Steals he wins:	\(\$13,600\)	\(\$ 0\)

Intersection of payoff tables

In a game, the best choices are interdependent:

(Nick: \(\$x\), Ibrahim: \(\$y\))	Ibrahim Splits	Ibrahim Steals
Nick Splits	\((\$6,800, \$6,800)\)	\((\$0, 13,600)\)
Nick Steals	\((\$13,600, \$0)\)	\((\$0, \$0)\)

Split or Steal Predictions

What do you think will happen in this game?

• Would you choose Split or Steal

• What would you say to your opponent?

Split or Steal: Nick and Ibrahim

Simultaneous-Move Games

What are Simultaneous-Move Games?

• Recall from Chapter 2: Simultaneous games occur when players must move without knowing their rivals’ choices.
• This includes cases where decisions are made at different times but without knowledge of others’ decisions.

Real-World Examples

• Soccer penalty kick: Striker and goalie must choose directions simultaneously, leading to strategic guessing.
• Television manufacturers: Decide on features and pricing without knowing competitors’ decisions.
• Voting: Voters cast ballots without knowing others’ choices.

Simultaneous vs Sequential Games

Simultaneous games

• neither know the other player’s actions when they act
• graphical representation: game matrix (aka game table, payoff table)
• solution method: best response, elimination of dominated strategies
• equlibrium concept: nash equilibrium

Sequential games

• prior actions are observable to later players
• graphical representation: game tree
• solution method: backwards induction (aka rollback)
• equlibrium concept: subgame perfect nash equilibrium (aka SPNE, rollback equilibrium)

The Strategic Form

A Strategic Form game is defined by:

• A set of players who have decisions

• A strategy set for each player which is the collection of strategies they choose from

• A payoff function which tells us how the player evaluates a strategy profile.

Tip

A strategy profile is a combination of strategies made by all players.

Depicting Simultaneous Games

• Simultaneous-move games are often depicted using payoff matrices or game tables.

Table 1: An example of a strategic form game

Row, Column	Left	Middle	Right
Top	(3,1)	(2,3)	(10,2)
High	(4,5)	(3,0)	(6,4)
Low	(2,2)	(5,4)	(12,3)
Bottom	(5,6)	(4,5)	(9,7)

• Row chooses a strategy, and Column does as well, simultaneously.

Strategy Sets in Table 1

In this game, each players’ complete strategy only contains one action each

• Player 1:

  • (Top), (High), (Low), or (Bottom)

Strategy Sets in Table 1

In this game, each players’ complete strategy only contains one action each

• Player 2:

  • (Left), (Middle), (Right)

• For now, we assume that these players can only choose one strategy at a time

  • I.e., not (Left, Right), (High, Low, Bottom), etc

  • In Chapter 7, we will relax this assumption by allowing for random mixing of strategies

Strategy Profiles in Table 1

There are twelve different strategy profiles possible in this game:

• {T,L}, {T,M}, {T,R}, {H,L}, {H,M}, {H,R}, {L,L}, {L,M}, {L,R}, {B,L}, {B,M}, {B,R}

each of which was represented by a cell in our game table.

Payoffs in Table Table 1

Notice how each cell has two numbers for the payoffs.

• For example, the first cell for the strategy profile { Top, Left }:

  • (3,1) means that the row player has a payoff of 3, the column player gets a payoff of 1

Big 10 Game

Table 2: Big 10 Game

Ducks , Buckeyes	Run	Pass	Blitz
Run	2, –2	5, –5	13, –13
Short Pass	6, –6	5.6, –5.6	10.5, –10.5
Medium Pass	6, –6	4.5, –4.5	1, –1
Long Pass	10, –10	3, –3	–2, 2

Notice that in this game, the payoffs in every cell add up to one

• This is what we mean by a zero-sum game

Nash Equilibrium

Solving Simultaneous Games

• Before we cover how to solve a game, we first need to discuss what a solution to a game even is.

• As with many types of problem in economics, we’re interested in finding an equilibrium:

  • equilibrium a situation which is “stable,” in that there is no incentive to change.

Nash Equilibrium

The most basic form of equilibrium in game theory is the Nash Equilibrium (NE), which can be described as:

• A strategy profile such that no player can obtain a larger payoff by unilaterally deviating

• A strategy profile such that no single player can make themselves better off by changing only their own strategy.

• A strategy profile such that, after the game is played, each player is satisfied that they could not have made a better decision

Nash Equilibrium

Nash Equilibrium

A list of strategies where each player’s strategy is the best response to the other’s best response strategy

• In other words, no player has any incentive to deviate in their strategy away from an equilibrium strategy
• It is central to the analysis of simultaneous-move games.
• Can occur in pure strategies (where players choose a single action) or mixed strategies (randomized choices).

Nash Equilibrium in Table 1

Let’s look at one specific strategy profile in this game:

	Left	Middle	Right
Top	(3,1)	(2,3)	(10,2)
High	(4,5)	(3,0)	(6,4)
Low	(2,2)	(5,4)	(12,3)
Bottom	(5,6)	(4,5)	(9,7)

Row chooses Low and Column chooses Middle

• Is this stable?

Nash Equilibrium in Table 1

Let’s look at one specific strategy profile in this game:

	Left	Middle	Right
Top	(3,1)	(2,3)	(10,2)
High	(4,5)	(3,0)	(6,4)
Low	(2,2)	(5,4)	(12,3)
Bottom	(5,6)	(4,5)	(9,7)

Row chooses Low and Column chooses Middle

• Given that Column plays Middle, is there any way for Row to get a higher payoff?

Nash Equilibrium in Table 1

Let’s look at one specific strategy profile in this game:

	Left	Middle	Right
Top	(3,1)	(2,3)	(10,2)
High	(4,5)	(3,0)	(6,4)
Low	(2,2)	(5,4)	(12,3)
Bottom	(5,6)	(4,5)	(9,7)

Row chooses Low and Column chooses Middle

• Given that Row plays Low, is there any way for Column to get a higher payoff?

Nash Equilibrium in Table 1

What is special about the strategy profile (Low, Middle)?

• It’s stable because neither player would have any incentive to change what they’re doing

• In other words, neither player has regrets over their choice given what the other player is doing

• However, notice that this doesn’t mean that this is the best that the players could do jointly

Other Strategy Profiles in Table 1

	Left	Middle	Right
Top	(3,1)	(2,3)	(10,2)
High	(4,5)	(3,0)	(6,4)
Low	(2,2)	(5,4)	(12,3)
Bottom	(5,6)	(4,5)	(9,7)

What about (Bottom, Right)?

Other Strategy Profiles in Table 1

	Left	Middle	Right
Top	(3,1)	(2,3)	(10,2)
High	(4,5)	(3,0)	(6,4)
Low	(2,2)	(5,4)	(12,3)
Bottom	(5,6)	(4,5)	(9,7)

Even though (Bottom, Right) is a Pareto improvement on (Low, Middle), it is unstable because it would be rational for Row player to change their strategy to Low to try to get the higher payoff of 12

Other Strategy Profiles in Table 1

	Left	Middle	Right
Top	(3,1)	(2,3)	(10,2)
High	(4,5)	(3,0)	(6,4)
Low	(2,2)	(5,4)	(12,3)
Bottom	(5,6)	(4,5)	(9,7)

What about (Top, Left)?

Table 1 with a tie in payoffs

	Left	Middle	Right
Top	(3,1)	(2,3)	(10,2)
High	(4,5)	(3,0)	(6,4)
Low	(2,2)	(5,4)	(12,3)
Bottom	(5,6)	(5,5)	(9,7)

A Nash equilibrium doesn’t require the equilibrium strategies to be strict improvements.

• (Low, Middle) is still a NE because the payoff of 5 to Row player is weakly (\(\geq\)) higher than the payoff of Bottom

Nash Equilibrium as a System of Beliefs

So a Nash Equilibrium requires players best responding to each other.

• But if a game is simultaneous, then players can’t observe what the other does

• So how can someone respond to something that hasn’t happened yet?

Nash Equilibrium as a System of Beliefs

How to make sense of best-responses in simultaneous games:

• Players might base their beliefs of the other players’ strategies on experience from playing similar games before

• You might also put yourself into your opponents’ shoes to think about what they might do

• You should also expect that they are thinking about what you’re thinking

  • and also that you would be thinking about what they are thinking about what you are thinking

The Battle of Wits

Let’s see an example of this logic in action:

Nash Equilibrium as a System of Beliefs

• Nash equilibrium assumes:
  1. Players form beliefs about the strategies of others.
  2. These beliefs must be correct.
  3. Each player then chooses their best response based on these beliefs.

Strategic Uncertainty

• Players are uncertain about others’ choices but form subjective beliefs about them.
• These beliefs are central to decision-making in Nash equilibrium.
• The equilibrium is found when every player’s belief about others’ actions is correct, and their response to these beliefs is optimal.

An Equivalent Definition of NE ¹

Nash Equilibrium

A set of strategies, one for each player, such that:

1. each player has correct beliefs about the strategies of the others and

2. the strategy of each is the best for herself, given her beliefs about the strategies of the others.

1. Dixit, pg. 98

Prisoners’ Dilemma

Prisoners’ Dilemma

To see the concept of Nash Equilibrium in action, we will start with the most famous example.

Prisoners’ Dilemma ¹

There are two criminals, who I like to call Guido and Luca.

• Guido and Luca have each committed murder, but the police have no way of proving it.
• However, the police can prove that they have lied on their taxes, which is a much lesser offense.
• The police get Guido and Luca into separate rooms so that they can’t talk, and offer them a choice: Testify against the other, or stay Quiet.
• If they Testify, they will receive lenient sentencing, and the police will be able to convict the other criminal.

1. as defined by Zachary Kiefer

Prisoner’s Dilemma

There are four different possibilities of what could happen:

• Both Keep Quiet: Police can only convict them of tax fraud, so Guido and Luca only serve 1 year in prison each
• Guido Testifies, but Luca Keeps Quiet: Guido gets 0 prison time for cooperating, Guido gets hit with a full 20 year sentence
• Luca Testifies, but Guido Keeps Quiet: Luca gets 0 prison time, Luca gets hit with a full 20 year sentence
• Both Testify: Both get convicted of murder, but get their prison sentences reduced to 10 years each because they cooperated

Prisoner’s Dilemma in Strategic Form

Guido, Luca	Testify	Keep Quiet
Testify	-10, -10	0, -20
Keep Quiet	-20, 0	-1, -1

What would you choose?

• Remember that you cannot communicate with your fellow prisoner

• How would you have to be that they won’t sell you out for you to choose to keep quiet?

Prisoners’ Dilemma

Let’s think through this from Guido’s perspective.

• Imagine that Luca keeps Quiet. This gives Guido a choice of two options: he can also keep Quiet, and get a short prison sentence, or he can Testify and get released. Obviously, Guido prefers to Testify.

• Now imagine that Luca Testifies. This again gives Guido a choice of two options: he can also keep Quiet, and get a long prison sentence, or he can Testify and get a medium sentence. Obviously, Guido still prefers to Testify.

• In other words, Guido prefers to Testify, no matter what Luca does.

• If we go through this logic from Luca’s perspective, we find that Luca also always prefers to Testify.

Nash Equilibrium in the Prisoners’ Dilemma

Although the outcome where both criminals keep quiet Pareto dominates the outcome where they both testify and serve 10 years, would it be stable?

• In the {Quiet, Quiet} strategy profile, either player could deviate to Testify and shorten their prison time.

• {Testify, Testify} is the only stable strategy profile, where no one could unilaterally deviate to decrease their sentence.

Paradox of the Prisoner’s Dilemma

• So, if Guido and Luca each act in their own self-interest, they will wind up each getting a medium prison sentence.
• This is despite the fact that, if they both kept Quiet, they could each get a short prison sentence instead.

Does this look familiar?

	Split	Steal
Split	\((\$6,800, \$6,800)\)	\((\$0, 13,600)\)
Steal	\((\$13,600, \$0)\)	\((\$0, \$0)\)

Although the numbers are different, the structure is the same.

Split or Steal is a Prisoners’ Dilemma

Watch what happens

start from 2:20

Discuss

• Did Nick change the structure of the game?

Prisoner’s Dilemma

Guido, Luca	Testify	Keep Quiet
Testify	-10,-10	0,-20
Keep Quiet	-20,0	-1,-1
False Confession	-20, 0	-20, -1

• What if we change the Prisoner’s Dilemma like this? What would you pick if you were Guido?

  1. Testify
  2. Keep Quiet
  3. False Confession

Prisoner’s Dilemma

Guido, Luca	Testify	Keep Quiet	False Confession
Testify	-10, -10	0, -20	0, -20
Keep Quiet	-20, 0	-1, -1	-1, -20
False Confession	-20, 0	-20, -1	-10, -10

• What about this third variation? What would you pick if you were Guido?

  1. Testify
  2. Keep Quiet
  3. False Confession

Rock Paper Scissors

One last question, and this one isn’t based on a Prisoner’s Dilemma…

Rock Paper Scissors

Lisa, Bart	R	P	S
R	0, 0	-1, 1	1, -1
P	1, -1	0, 0	-1, 1
S	-1, 1	1, -1	0,0

• This game models a game of Rock-Paper-Scissors. If you are Lisa, which strategy will you choose?

  1. R(ock)
  2. P(aper)
  3. S(cissors)

Obviously-Wrong Strategies

• The first three of those games contained strategies that were obviously bad choices.

• Rock-Paper-Scissors did not.

• One of the simplest things you can do with a strategic-form game is to start by finding and eliminating (ruling out) the strategies which are obviously bad.

• In some games, this can even be enough to identify the Nash Equilibrium!

Dominance

Dominance and Strategy Elimination

• A dominant strategy is one that performs best no matter what others do.
• A strategy is dominated if there is another strategy which is better no matter what the other player does.

Elimination of dominated strategies can simplify finding equilibria by removing strategies

• Dominated strategies will never by played by a rational agent

The Problem of Finding Nash Equilibria

• When we first discussed NE, we found them by checking all of the strategy profiles in the game to see which of them were stable.

• This is easy for a 2x2 game or even a 3x3 game like we’ve seen

  • but it gets much more time-consuming in games with more players and more strategies per player

• We can make it easier to find NEs with a few useful shortcuts

Strict Dominance

• A strategy is said to be strictly dominated if there is some other strategy, in the same player’s strategy set, which provides that player a higher payoff, no matter what strategies the other players pick.

  • Another way to phrase it is that a strategy is strictly dominated if some other strategy is a better alternative for the player, no matter what other players do.

Strict Dominance

Guido, Luca	Testify	Keep Quiet
Testify	-10, -10	0, -20
Keep Quiet	-20, 0	-1, -1

• Both Guido and Luca prefer to Testify, no matter whether the other player chose:

  • this means that Quiet is [strictly dominated]{,hi}, by Testify for both players.

Strict Dominance

Guido, Luca	Testify	Keep Quiet
Testify	-10, -10	0, -20
Keep Quiet	-20, 0	-1, -1

• It is not rational to play a strictly dominated strategy

  • So we can immediately deduce that neither player would play Quiet, and the only remaining strategy profile is (Testify, Testify).

One Player has a Dominant Strategy

In the Prisoners’ Dilemma, both players have each have one dominant strategy, which made finding the NE easy.

CONGRESS, FEDERAL RESERVE	Low interest rates	High interest rates
Budget balance	3,4	4,1
Budget deficit	1,3	2,2

• What if only one player has a dominant strategy?

One Player has a Dominant Strategy

CONGRESS, FEDERAL RESERVE	Low interest rates	High interest rates
Budget balance	3,4	1,3
Budget deficit	4,1	2,2

• The Fed would like to set low interest rates, but only if Congress keeps the budget under control

  • Neither Low nor High interest rates are a dominant strategy

One Player has a Dominant Strategy

CONGRESS, FEDERAL RESERVE	Low interest rates	High interest rates
Budget balance	3,4	1,3
Budget deficit	4,1	2,2

• But Congress wants to run a budget deficit no matter what interest rates are

Finding NEs by Elimination

• If all but one of each player’s strategies can be eliminated like this (leaving only a single strategy profile), then the remaining strategy profile is a NE.
  • A strictly dominated strategy can never be part of a NE.
• However, it’s rare that a player has one strategy which strictly dominates all of their others from the very start, as in the Prisoner’s Dilemma. (This is called a strictly dominant strategy.)
• Even if a player doesn’t have a strictly dominant strategy, we can still sometimes use elimination to find a NE, by using a process called Iterated Elimination of Strictly Dominated Strategies (IESDS).

Commonly Known Rationality

• Let’s assume that, not only is every player rational, they all know that the other players are rational too.
• This means that players can deduce which strategies the other players would never play.
• And if a player can eliminate another player’s strategy, it may reveal additional strictly dominated strategies that can be eliminated.

Eliminating Strictly Dominated Strategies

\(P_1\), \(P_2\)	\(a\)	\(b\)	\(c\)
\(A\)	1, 1	2, 2	3, 3
\(B\)	2, 0	3, 1	4, 2
\(C\)	3, 1	2, 2	1, 1

• In the game table above, there are no strictly dominant strategies.
  • For Player 1, A is strictly dominated by B, but C is neither dominant nor dominated.
  • And for Player 2, a is strictly dominated by b, but c is also neither dominant nor dominated.

Eliminating Strictly Dominated Strategies

\(P_1\), \(P_2\)	\(a\)	\(b\)	\(c\)
\(A\)	1, 1	2, 2	3, 3
\(B\)	2, 0	3, 1	4, 2
\(C\)	3, 1	2, 2	1, 1

• However, the assumption of Commonly Known Rationality allows Player 1 to deduce that Player 2 would never play a.

Eliminating Strictly Dominated Strategies

\(P_1\), \(P_2\)	\(a\)	\(b\)	\(c\)
\(A\)	1, 1	2, 2	3, 3
\(B\)	2, 0	3, 1	4, 2
\(C\)	3, 1	2, 2	1, 1

• Player 1 can eliminate a, just like we did—and once they do, C is strictly dominated by B.

Eliminating Strictly Dominated Strategies

\(P_1\), \(P_2\)	\(a\)	\(b\)	\(c\)
\(A\)	1, 1	2, 2	3, 3
\(B\)	2, 0	3, 1	4, 2
\(C\)	3, 1	2, 2	1, 1

• Player 2 can deduce all of this
  • and once they eliminate A, a, and C, b is strictly dominated by c.
• This leaves us one strategy per player, and so the NE here is (B, c).

Eliminating Strictly Dominated Strategies

\(P_1\), \(P_2\)	\(a\)	\(b\)	\(c\)
\(A\)	1, 1	2, 2	3, 3
\(B\)	2, 0	3, 1	4, 2
\(C\)	3, 1	2, 2	1, 1

• This leaves us one strategy per player, and so the NE here is [(B, c)]{,hi}.

Order Doesn’t Matter

• In IESDS, the order in which you eliminate strategies doesn’t matter. You’ll get the same result no matter how you do it—as long as you keep going to the end.

Order Doesn’t Matter

Consider this even larger game:

\(P_1\), \(P_2\)	\(a\)	\(b\)	\(c\)	\(d\)	\(e\)
\(A\)	1, 1	2, 2	2, 2	2, 1	4, 1
\(B\)	1, 3	1, 3	2, 2	2, 3	3, 2
\(C\)	1, 2	2, 4	1, 3	2, 3	1, 3
\(D\)	3, 2	2, 3	1, 4	2, 2	1, 2
\(E\)	2, 1	3, 2	3, 2	3, 3	4, 1

Order Doesn’t Matter (start w/ \(P_1\))

Suppose we start by looking for Player 1’s strictly dominated strategies:

\(P_1\), \(P_2\)	\(a\)	\(b\)	\(c\)	\(d\)	\(e\)
\(A\)	1, 1	2, 2	2, 2	2, 1	4, 1
\(B\)	1, 3	1, 3	2, 2	2, 3	3, 2
\(C\)	1, 2	2, 4	1, 3	2, 3	1, 3
\(D\)	3, 2	2, 3	1, 4	2, 2	1, 2
\(E\)	2, 1	3, 2	3, 2	3, 3	4, 1

Order Doesn’t Matter (start w/ \(P_1\))

Suppose we start by looking for Player 1’s strictly dominated strategies:

\(P_1\), \(P_2\)	\(a\)	\(b\)	\(c\)	\(d\)	\(e\)
\(A\)	1, 1	2, 2	2, 2	2, 1	4, 1
\(B\)	1, 3	1, 3	2, 2	2, 3	3, 2
\(C\)	1, 2	2, 4	1, 3	2, 3	1, 3
\(D\)	3, 2	2, 3	1, 4	2, 2	1, 2
\(E\)	2, 1	3, 2	3, 2	3, 3	4, 1

• \(B\) and \(C\) are strictly dominated by \(E\)

Order Doesn’t Matter (start w/ \(P_1\))

\(P_1\), \(P_2\)	\(a\)	\(b\)	\(c\)	\(d\)	\(e\)
\(A\)	1, 1	2, 2	2, 2	2, 1	4, 1
\(B\)	1, 3	1, 3	2, 2	2, 3	3, 2
\(C\)	1, 2	2, 4	1, 3	2, 3	1, 3
\(D\)	3, 2	2, 3	1, 4	2, 2	1, 2
\(E\)	2, 1	3, 2	3, 2	3, 3	4, 1

• Are there any newly strictly dominated strategies after eliminating \(B\) and \(C\)?

Order Doesn’t Matter (start w/ \(P_1\))

\(P_1\), \(P_2\)	\(a\)	\(b\)	\(c\)	\(d\)	\(e\)
\(A\)	1, 1	2, 2	2, 2	2, 1	4, 1
\(B\)	1, 3	1, 3	2, 2	2, 3	3, 2
\(C\)	1, 2	2, 4	1, 3	2, 3	1, 3
\(D\)	3, 2	2, 3	1, 4	2, 2	1, 2
\(E\)	2, 1	3, 2	3, 2	3, 3	4, 1

• After eliminating \(B\) and \(C\), now \(a\) and \(e\) are strictly dominated by \(b\) for Player 2

• This is conditional on Player 2 knowing that if Player 1 is rational, they would never play \(B\) or \(C\)

Order Doesn’t Matter (start w/ \(P_1\))

\(P_1\), \(P_2\)	\(a\)	\(b\)	\(c\)	\(d\)	\(e\)
\(A\)	1, 1	2, 2	2, 2	2, 1	4, 1
\(B\)	1, 3	1, 3	2, 2	2, 3	3, 2
\(C\)	1, 2	2, 4	1, 3	2, 3	1, 3
\(D\)	3, 2	2, 3	1, 4	2, 2	1, 2
\(E\)	2, 1	3, 2	3, 2	3, 3	4, 1

• Now \(A\) and \(D\) are strictly dominated by \(E\) for Player 1

Order Doesn’t Matter (start w/ \(P_1\))

\(P_1\), \(P_2\)	\(a\)	\(b\)	\(c\)	\(d\)	\(e\)
\(A\)	1, 1	2, 2	2, 2	2, 1	4, 1
\(B\)	1, 3	1, 3	2, 2	2, 3	3, 2
\(C\)	1, 2	2, 4	1, 3	2, 3	1, 3
\(D\)	3, 2	2, 3	1, 4	2, 2	1, 2
\(E\)	2, 1	3, 2	3, 2	3, 3	4, 1

• And finally, \(b\) and \(c\) are strictly dominated by \(d\)

Order Doesn’t Matter (start w/ \(P_1\))

\(P_1\), \(P_2\)	\(a\)	\(b\)	\(c\)	\(d\)	\(e\)
\(A\)	1, 1	2, 2	2, 2	2, 1	4, 1
\(B\)	1, 3	1, 3	2, 2	2, 3	3, 2
\(C\)	1, 2	2, 4	1, 3	2, 3	1, 3
\(D\)	3, 2	2, 3	1, 4	2, 2	1, 2
\(E\)	2, 1	3, 2	3, 2	3, 3	4, 1

So we are left with a single strategy profile:

• (\(E\) , \(d\))

Order Doesn’t Matter (start w/ \(P_1\))

\(P_1\), \(P_2\)	\(a\)	\(b\)	\(c\)	\(d\)	\(e\)
\(A\)	1, 1	2, 2	2, 2	2, 1	4, 1
\(B\)	1, 3	1, 3	2, 2	2, 3	3, 2
\(C\)	1, 2	2, 4	1, 3	2, 3	1, 3
\(D\)	3, 2	2, 3	1, 4	2, 2	1, 2
\(E\)	2, 1	3, 2	3, 2	3, 3	4, 1

• is this a NE?

  • is there any way either player could unilateraly deviate to a higher payoff?

Order Doesn’t Matter (start w/ \(P_2\))

Now what happens if we reset and start instead by looking for Player 2’s strictly dominated strategies?

\(P_1\), \(P_2\)	\(a\)	\(b\)	\(c\)	\(d\)	\(e\)
\(A\)	1, 1	2, 2	2, 2	2, 1	4, 1
\(B\)	1, 3	1, 3	2, 2	2, 3	3, 2
\(C\)	1, 2	2, 4	1, 3	2, 3	1, 3
\(D\)	3, 2	2, 3	1, 4	2, 2	1, 2
\(E\)	2, 1	3, 2	3, 2	3, 3	4, 1

Order Doesn’t Matter (start w/ \(P_2\))

\(P_1\), \(P_2\)	\(a\)	\(b\)	\(c\)	\(d\)	\(e\)
\(A\)	1, 1	2, 2	2, 2	2, 1	4, 1
\(B\)	1, 3	1, 3	2, 2	2, 3	3, 2
\(C\)	1, 2	2, 4	1, 3	2, 3	1, 3
\(D\)	3, 2	2, 3	1, 4	2, 2	1, 2
\(E\)	2, 1	3, 2	3, 2	3, 3	4, 1

• \(a\) and \(e\) are strictly dominated by \(b\)

Order Doesn’t Matter (start w/ \(P_2\))

\(P_1\), \(P_2\)	\(a\)	\(b\)	\(c\)	\(d\)	\(e\)
\(A\)	1, 1	2, 2	2, 2	2, 1	4, 1
\(B\)	1, 3	1, 3	2, 2	2, 3	3, 2
\(C\)	1, 2	2, 4	1, 3	2, 3	1, 3
\(D\)	3, 2	2, 3	1, 4	2, 2	1, 2
\(E\)	2, 1	3, 2	3, 2	3, 3	4, 1

• \(A\), \(B\), \(C\), and \(D\) are now strictly dominated by \(E\)

Order Doesn’t Matter (start w/ \(P_2\))

\(P_1\), \(P_2\)	\(a\)	\(b\)	\(c\)	\(d\)	\(e\)
\(A\)	1, 1	2, 2	2, 2	2, 1	4, 1
\(B\)	1, 3	1, 3	2, 2	2, 3	3, 2
\(C\)	1, 2	2, 4	1, 3	2, 3	1, 3
\(D\)	3, 2	2, 3	1, 4	2, 2	1, 2
\(E\)	2, 1	3, 2	3, 2	3, 3	4, 1

• \(b\) and \(c\) are now strictly dominated by \(d\)

Order Doesn’t Matter (start w/ \(P_2\))

\(P_1\), \(P_2\)	\(a\)	\(b\)	\(c\)	\(d\)	\(e\)
\(A\)	1, 1	2, 2	2, 2	2, 1	4, 1
\(B\)	1, 3	1, 3	2, 2	2, 3	3, 2
\(C\)	1, 2	2, 4	1, 3	2, 3	1, 3
\(D\)	3, 2	2, 3	1, 4	2, 2	1, 2
\(E\)	2, 1	3, 2	3, 2	3, 3	4, 1

Once again we’re left with the same strategy profile:

• (\(E\) , \(d\))

IESDS in a Nutshell

• The process of IESDS can be summed up in three steps:

  1. Search for a strictly dominated strategy belonging to any player. If none exists, stop here: IESDS is completed.
  2. Eliminate (cross out) that strategy. Optionally, re-draw the game table without the eliminated strategy.
  3. Return to step 1.

IESDS Example:

\(P_1\), \(P_2\)	x	y	z
X	1,3	2,2	3,2
Y	2,2	2,2	4,3
Z	1,1	0,2	1,1

Which strategy is strictly dominated?

IESDS Example:

\(P_1\), \(P_2\)	x	y	z
X	1,3	2,2	3,2
Y	2,2	2,2	4,3
Z	1,1	0,2	1,1

What does IESDS tell you about the NE of this game?

Failures of Elimination Methods

• There’s no guarantee that any particular game will contain strictly dominated strategies
• Even when there are strategies we can eliminate, there may not be enough of them to find a NE just by elimination.
• So why bother with this?
• Even if elimination doesn’t immediately identify a NE, it can still be helpful to simplify the game before trying other methods.
• Simplifying by elimination is especially useful when dealing with mixed strategies

Best-Response Analysis

Limits of IESDS

Not every simultaneous-move strategy can by only eliminating dominated strategies

Can we still find NE in those games?

• Yes! We just need to remember the best-responding part of our definition of NE

Best-Responding

• In finding Nash equilibrium, best-response analysis is key.
• Each player chooses their best action in response to the strategies of others.
• This iterative process helps in solving games with multiple strategies.

Best Responses

• If you know what strategy the other player will choose, then you can easily figure out what your best option (or options) are.
• Of course, you don’t actually know what the other player will do when you choose your strategy in a game like this—but thinking about the game this way makes it easy to find NEs.
• A strategy \(s_i\) is a best response to another player’s strategy \(s_{-i}\) if and only if it provides the highest payoff possible when the other player chooses \(s_{-i}\).

Some Notes on Best Responses

• The phrase “to another player’s strategy” in the definition of a best response is important.
• A strategy can only be a best response to some strategy of the other player. There is no such thing as a strategy which is just “a best response.”
• When dealing with strategic-form games with game tables, there is always at least one best response to another player’s strategy—and there may be multiple, if there is more than one strategy which provides the best payoff.

Best Responses in Practice

• It’s easy to depict best responses in a game table: we can go through each strategy in the game, and mark each strategy which is a best response to them.
• We do this by marking the payoffs—however, it’s important to understand that it’s not the payoff itself which is a best response–we’re just using them as a convenient way to depict where the best responses are.
  • In other words: don’t try to describe the best response using the payoff. The best response is always the strategy.

Best Response Analysis

Let’s revisit this game from the beginning of the lecture:

Row, Column	Left	Middle	Right
Top	3, 1	2, 3	10, 2
High	4, 5	3, 0	6, 4
Low	2, 2	5, 4	12, 3
Bottom	5, 6	4, 5	9, 7

• We’ll break down the best responses one step at a time

Best Response Analysis

Suppose you’re the Row player:

Row	Left
Top	3
High	4
Low	2
Bottom	5

What would you do if you knew Column will play Left?

Best Response Analysis

Row	Left
Top	3
High	4
Low	2
Bottom	5

We’ll underline this payoff as Bottom is the best response to Left

Best Response Analysis

Row	Left	Middle
Top	3	2
High	4	3
Low	2	5
Bottom	5	4

Also, Low is the best response to Middle

Best Response Analysis

Row	Left	Middle	Right
Top	3	2	10
High	4	3	6
Low	2	5	12
Bottom	5	4	9

And Low is also the best response to Right

Best Response Analysis

Row, Column	Left	Middle	Right
Top	3, 1	2, 3	10, 2
High	4	3	6
Low	2	5	12
Bottom	5	4	9

For Column, Middle is a best response to Top

Best Response Analysis

Row, Column	Left	Middle	Right
Top	3, 1	2, 3	10, 2
High	4, 5	3, 0	6, 6
Low	2	5	12
Bottom	5	4	9

Left is a best response to High

Best Response Analysis

Row, Column	Left	Middle	Right
Top	3, 1	2, 3	10, 2
High	4, 5	3, 0	6, 6
Low	2, 2	5, 4	12, 3
Bottom	5	4	9

Middle is a best response to Low

Best Response Analysis

Row, Column	Left	Middle	Right
Top	3, 1	2, 3	10, 2
High	4, 5	3, 0	6, 6
Low	2, 2	5, 4	12, 3
Bottom	5, 6	4, 5	9, 7

And Right is a best response to Bottom

Best Response Analysis

Row, Column	Left	Middle	Right
Top	3, 1	2, 3	10, 2
High	4, 5	3, 0	6, 6
Low	2, 2	5, 4	12, 3
Bottom	5, 6	4, 5	9, 7

The NE of this game is where the best responses intersect:

• (Low, Middle)

Three-Player Simultaneous Games

Let’s create a modified version of the three-roommate planting game:

• Now instead of observing who contributed to the garden already in certain order, suppose that Emily, Nina, and Talia have to choose simultaneously

• Also instead of the garden either dying or surviving, we’ll add more detailed levels of quality depending on who contributed

Simultaneous Planting Game

What are Emily’s preferences in this modified game? ¹

Outcome	Payoff
Nina and Talia contribute, Emily doesn’t	6
Everyone contributes	5
Either Nina or Talia contribute, Emily doesn’t	4
Either Nina or Talia contribute, and Emily does	3
No-one contributes	2
Emily is the only one to contribute	1

1. this is a symmetric game, so Nina and Talia will have equivalent rankings

Simultaneous Planting Game

Talia chooses:

Contribute

Emily, Nina	Contribute	Don’t
Contribute	5, 5, 5	3, 6, 3
Don’t	6, 3, 3	4, 4, 1

Don’t Contribute

Emily, Nina	Contribute	Don’t
Contribute	3, 3, 6	1, 4, 4
Don’t	4, 1, 4	2, 2, 2

• Can you find any dominant strategies?

Simultaneous Planting Game

Talia chooses:

Contribute

Emily, Nina	Contribute	Don’t
Contribute	5, 5, 5	3, 6, 3
Don’t	6, 3, 3	4, 4, 1

Don’t Contribute

Emily, Nina	Contribute	Don’t
Contribute	3, 3, 6	1, 4, 4
Don’t	4, 1, 4	2, 2, 2

Nash Equilibrium from Best Responses

Recall the various definitions of a Nash Equilibrium: * a strategy profile such that [no player can obtain a larger payoff by unilaterally deviating[{.hi}] (changing only their own strategy). * A strategy profile such that no single player can make themselves better off by changing only their own strategy. * A strategy profile such that, after the game is played, each player is satisfied that they could not have made a better decision.

• Another definition we can use now is “A strategy profile such that each player’s strategy is a best responses to the other player’s strategy.”
• “Playing a best response” is equivalent to “cannot obtain a larger payoff by unilaterally deviating,” or any of the other ways to describe this condition.

Classifying Games Based on NEs

• Now that we’ve talked about several ways to find a game’s NEs, we can start to talk about classifying games using them.

Multiple Equilibria in Pure Strategies

Games with Multiple Equilibria

So we’ve seen games with only one unique NE

• But this doesn’t have to be the case

• Next we will see games with multiple possible strategy profiles which are stable.

Coordination Games

I agreed to meet my friend for coffee, but we didn’t decide on which cafe and now my phone is dead:

Dante, Jose	Starbucks	Peet’s
Starbucks	1, 1	0, 0
Peet’s	0, 0	1, 1

Coordination Games

Dante, Jose	Starbucks	Peet’s
Starbucks	1, 1	0, 0
Peet’s	0, 0	1, 1

• There are two NE’s where we are both happy

• The problem is getting there

• This type of game is called a coordination game

Coordination Games

Even when there is an obviously preferred NE for both players, it may not be guaranteed that we will get there:

Dante, Jose	Starbucks	Roma
Starbucks	1, 1	0, 0
Roma	0, 0	2, 2

• Depending on the set of beliefs held by the players, both strategy profiles could be theoretically stable

• If for some reason we started out in a (Starbucks, Starbucks) equilibrium, there would be no incentive for either of us to deviate without communicating our intent to each other

Coordination Games

What should happen when different players have different preferences over NE?

Dante, Jose	T4	Metropol
T4	2, 1	0, 0
Metropol	0, 0	1, 2

• If Dante tries to be nice and go to Metropol, what if Jose does the same thing and goes to T4?
• In chapter 10, with repeated games, there may be opportunities for us to alternate between each of our favorites

The Deer Hunt Game

Two cavemen, Igg and Ogg, each decide whether to hunt Deer or Rabbit.

• If both hunt Deer, they can work together to bring one down and have enough meat to share with leftovers

  • Each earn payoffs of 2

• If instead the other goes off chasing a rabbit, the cameman hunting deer won’t be able to catch any meat, but the one hunting rabbit will get just enough to feed themselves

  • Unsuccessful Deer hunter gets 0, rabbit hunter gets 1

• Or both cavemen could just decide to go off and hunt rabbits by themselves, ensuring they get can feed themselves

  • Both get 1

The Deer Hunt Game

Igg, Ogg	Deer	Rabbit
Deer	2, 2	0, 1
Rabbit	1, 0	1, 1

• Are there any strictly dominated strategies?
• What about best responses?

The Deer Hunt Game

Here are Igg’s best responses…

Igg, Ogg	Deer	Rabbit
Deer	2, 2	0, 1
Rabbit	1, 0	1, 1

The Deer Hunt Game

And here are Ogg’s best responses…

Igg, Ogg	Deer	Rabbit
Deer	2, 2	0, 1
Rabbit	1, 0	1, 1

The Game of Chicken

James, Dean	Swerve (Chicken)	Straight (Tough)
Swerve	0, 0	-1, 1
Straight	1, -1	-2, -2

Suppose that James and Dean are driving headlong towards each other, but neither one wants to be the ‘chicken’ by swerving out of the way

• What are the best respones?

The Game of Chicken

James, Dean	Swerve (Chicken)	Straight (Tough)
Swerve	0, 0	-1, 1
Straight	1, -1	-2, -2

• There are two NE of this game

• But how can we decide which one will happen?

• Chapter 9 will look at potential commitment devices either player could use to try to achieve their preferred NE

Another Example: When Elimination Does Nothing

\(P_1\), \(P_2\)	\(a\)	\(b\)	\(c\)	\(d\)
\(A\)	1, 1	2, 2	2, 2	2, 1
\(B\)	1, 3	1, 3	2, 2	2, 3
\(C\)	1, 2	2, 4	1, 3	2, 3
\(D\)	3, 2	2, 3	1, 4	2, 2

• Can you find any strictly dominated strategies?
• Can you find any NE?

Classifying NEs

• Nash Equilibria may be either strict or weak.
• A Nash Equilibrium is strict if and only if each player would receive a (strictly) smaller payoff by changing their own strategy.
• If a Nash Equilibrium is not strict—meaning that at least one player could change their own strategy and receive an equal (but not larger) payoff
  • it is [weak]{hi}.

Intuition on Strict vs. Weak Equilibria

• In any Nash Equilibrium, no player has a reason to change their own strategy—they cannot get a higher payoff this way.
• Strict Nash Equilibria go a little further: not only does no player have a reason to change their own strategy, they also have a reason not to, because any other strategy would provide them a worse payoff.
• If a Nash Equilibrium is weak, it means that some player could change their strategy, and get exactly the same payoff they already were. They have no reason to do this, but also no reason not to.
• We can also say that a strict Nash Equilibrium is one where each player is playing a strategy which is a unique best response to the strategies chosen by other players.

Deer Hunt: Strict Nash Equilibria

Igg, Ogg	\(Deer\)	\(Rabbit\)
\(Deer\)	2, 2	0,1
\(Rabbit\)	1,0	1, 1

• Here, note that at each Nash Equilibrium, each player has no other strategy providing the same payoff.

• This is a strict Nash Equilibrium.

Weak Nash Equilibria Example

Eleanor, Michael	\(Swerve\)	\(Straight\)
\(Swerve\)	1, 1	1, 1
\(Straight\)	1, 1	0, 0

• However, in this game, each Nash Equilibrium features at least one player who could still get the same payoff if they change their strategy.

Prisoners’ Dilemmas

Guido, Luca	\(Testify\)	\(Keep~Quiet\)
\(Testify\)	-10, -10	0, -20
\(Keep~Quiet\)	-20, 0	-1, -1

In general, a Prisoner’s Dilemma is any game in which:

• The players have the same strategies, A and B.
• A strictly dominates B, making (A, A) the only NE.
• But (B, B) is better for both players than (A, A). :::

Coordination Games

• A coordination game is a game in which the players all have the same strategy sets, and the NEs are all of the strategy profiles where the players choose the same strategy.
• The Deer Hunt is a coordination game.

Other Coordination Games

\(P_1\), \(P_2\)	A	B	C
A	1, 1	1, 0	3, 0
B	0, 1	2, 2	2, 1
B	0, 3	1, 2	4, 4

Anti-Coordination Games

• An anti-coordination game is a game in which the players all have the same strategy sets, but the NEs are all of the strategy profiles where the players choose different strategies.

James, Dean	Swerve (Chicken)	Straight (Tough)
Swerve	0, 0	-1, 1
Straight	1, -1	-2, -2

Symmetric Games

• A symmetric game is a game which is indifferent to an exchange of players in other words, a game where the players are interchangeable.
• Consider the Prisoner’s Dilemma: if we swap the players’ names, and their positions in the game table, and the order of their payoffs, we get the same game that we started with.

Symmetric Games

• A two-player game with a game table is symmetric if:
  • The players have the same strategy sets.
  • In the on-diagonal cells of the game table, the players receive equal payoffs.
  • In the “mirrored” off-diagonal cells of the game table, the players’ payoffs are reversed.
• Of the games we’ve just looked at, the Deer Hunt, the 3x3 coordination game, and Chicken are symmetric games. Bach and Stravinsky is not.

Other Elimination Methods: Weakly Dominated Strategies

Weakly Dominated Strategies

• It is also possible to find Nash Equilibria by eliminating weakly dominated strategies

  • (strategies for which there is an alternative that is never worse, and sometimes better).

Weakly Dominated Strategies

We will not spend a lot of time on this method, because it has two serious flaws:

1. It is possible for a Nash Equilibrium to involve playing a weakly dominated strategy.

• IEWDS may therefore eliminate actual Nash Equilibria.

2. Unlike IESDS, the order in which you eliminate weakly dominated strategies matters:

• you may get different results from different orders of elimination.

Example: Why Weak Dominance is Not Useful

In the game below, B is weakly dominated for both players.

Player 1, Player 2	\(A\)	\(B\)
\(A\)	\(2,2\)	\(1,1\)
\(B\)	\(1,1\)	\(1,1\)

• If we eliminate the weakly dominated strategy for both players, then the only remaining strategy profile is (A, A)

  • this is a Nash equilibrium.

• However, (B, B) is also a Nash equilibrium:

  • both players get payoff 1, and neither can improve that payoff by changing their own strategy.

  • We failed to find this equilibrium by eliminating weakly dominated strategies.

Another Example: Why Weak Dominance is Not Useful

In the game below, M and R are weakly dominated.

Player 1, Player 2	\(L\)	\(M\)	\(R\)
\(T\)	0, 1	1, 0	0, 0
\(B\)	0, 0	0, 0	1, 0

• If we begin by eliminating R, then afterwards, M and B are both weakly dominated, and we would eliminate them, leaving only (T, L).

Another Example: Why Weak Dominance is Not Useful

Player 1, Player 2	\(L\)	\(M\)	\(R\)
\(T\)	0, 1	1, 0	0, 0
\(B\)	0, 0	0, 0	1, 0

• However, if we begin by eliminating M, then T and R are both weakly dominated, and if we eliminate them, we are left with only (B, L).

• Not only does the outcome of IEWDS depend on what we eliminate first, it still fails to find a third Nash equilibrium, which is (B, R).

Other Elimination Methods: Non-Best-Responses

• Generally speaking, we can eliminate any strategy which is not rational to play in a NE.
• It’s never rational to play strictly dominated strategies, but it’s sometimes rational to play weakly dominated strategies.
• There are other categories of non-rational strategies:
• A strategy is a non-best-response or non-rationalizable strategy if and only if, regardless of what the other players choose, it never provides the best possible payoff.
• I like to describe non-rationalizable strategies as strategies that you’d have to be crazy to think were a good idea, i.e. you can’t rationalize playing them.

Strictly Dominated vs. Non-Best-Response

• Non-rationalizability is very similar to strict dominance, but here’s the difference:
• Strict dominance is pairwise: a strategy \(s\) dominates another, \(s'\), if \(s\) specifically always gives a better payoff than \(s'\).
• Non-rationalizability is a property of a single strategy: for a strategy to be non-rationalizable, it means that there is always some option that gives a better payoff—but the better option doesn’t always have to be the same strategy.
• To put it another way: strategy \(s\) is not a best response if there is always some strategy* which is better. To be strictly dominated, there must be one particular strategy** which is always better.

Example: Non-Best-Responses

In the game below, there are no strictly dominated strategies, meaning that IESDS will not do anything to simplify it.

		P\(_2\)
		\(a\)	\(b\)	\(c\)
3-5 *P\(_1\)	\(A\)	0, 4	1, 2	3, 3
3-5	\(B\)	1, 2	0, 3	1, 4
3-5	\(C\)	3, 3	1, 2	0, 1
3-5

• However, we can see that B is non-rationalizable for Player 1: regardless of whether Player 2 chooses a, b, or c, Player 1 is better off playing either A or C.

• Also, b is non-rationalizable for Player 2.

Another Example: Non-Best-Responses

		P\(_2\)
		\(a\)	\(b\)	\(c\)	\(d\)
3-6 *P\(_1\)	\(A\)	1, 0	2, 1	3, 1	4, 2
3-6	\(B\)	3, 1	2, 2	2, 0	3, 0
3-6	\(C\)	3, 1	4, 0	1, 0	2, 0
3-6	\(D\)	4, 2	3, 0	2, 1	1, 1
3-6

Iterated Elimination of Non-Best-Responses

• It is not rational to play a non-best-response strategy in a pure-strategy context. (As we’ll see much later, it’s more complicated in a mixed-strategy context).
• Any strictly dominated strategy is also not a best response, but not all NBR strategies are strictly dominated.
• Because of this, if we eliminate non-best-responses, using the same steps as IESDS, this process will always eliminate the same strategies, and it may eliminate even more!
• This process is, naturally, called Iterative Elimination of Non-Best-Responses (IENBR).

Example: IENBR

Returning to this same example, we can start by eliminating B.

		P\(_2\)
		\(a\)	\(b\)	\(c\)
3-5 *P\(_1\)	\(A\)	0, 4	1, 2	3, 3
3-5	\(B\)	1, 2	0, 3	1, 4
3-5	\(C\)	3, 3	1, 2	0, 1
3-5

• Once we do this, we can now see that b and c are non-rationalizable, and eliminate them.
• Finally, we can see that B is non-rationalizable and eliminate it, leaving only the strategy profile (C, a), which is the Nash equilibrium of this game.

Another Example: IENBR

		P\(_2\)
		\(a\)	\(b\)	\(c\)	\(d\)
3-6 *P\(_1\)	\(A\)	1, 0	2, 1	3, 1	4, 2
3-6	\(B\)	3, 1	2, 2	2, 0	3, 0
3-6	\(C\)	3, 1	4, 0	1, 0	2, 0
3-6	\(D\)	4, 2	3, 0	2, 1	1, 1
3-6

iClicker Q4

		P\(_2\)
		\(x\)	\(y\)	\(z\)
3-5 *P\(_1\)	\(X\)	\(1,3\)	\(2,2\)	\(3,2\)
3-5	\(Y\)	\(2,2\)	\(2,2\)	\(4,3\)
3-5	\(Z\)	\(1,1\)	\(0,2\)	\(1,1\)
3-5

• After eliminating Z, which is strictly dominated, what strategy is a non-best-response?
  1. X
  2. Y
  3. Z
  4. y
  5. z

iClicker Q5

		P\(_2\)
		\(x\)	\(y\)	\(z\)
3-5 *P\(_1\)	\(X\)	\(1,3\)	\(2,2\)	\(3,2\)
3-5	\(Y\)	\(2,2\)	\(2,2\)	\(4,3\)
3-5	\(Z\)	\(1,1\)	\(0,2\)	\(1,1\)
3-5

• If we complete IENBR on this game table, what strategy profile is left at the end?
  1. (X, x)
  2. (X, z)
  3. (Y, x)
  4. (Y, z)
  5. There is more than one strategy profile left.

Failures of Elimination Methods

• There’s no guarantee that any particular game will contain strategies that are either strictly dominated or non-rationalizable.
• Even when there are strategies we can eliminate, there may not be enough of them to find a NE just by elimination.
• So why bother with this?
• Even if elimination doesn’t immediately identify a NE, it can still be helpful to simplify the game before trying other methods.
• Simplifying by elimination is especially useful when dealing with mixed strategies—which we’ll get to after the midterm.

Another Example: When Elimination Does Nothing

• This game has absolutely no strategies that can be eliminated: none are strictly dominated or non-rationalizable.

• We can still find the NEs (of which there are quite a few) using best responses:

		P\(_2\)
		\(a\)	\(b\)	\(c\)	\(d\)
3-6 *P\(_1\)	\(A\)	1, 1	2, 2	2, 2	2, 1
3-6	\(B\)	1, 3	1, 3	2, 2	2, 3
3-6	\(C\)	1, 2	2, 4	1, 3	2, 3
3-6	\(D\)	3, 2	2, 3	1, 4	2, 2
3-6

iClicker Q4

		P\(_2\)
		\(X\)	\(Y\)	\(Z\)
3-5 *P\(_1\)	\(A\)	3, 3	2, 2	1, 1
3-5	\(B\)	4, 2	1, 1	2, 2
3-5	\(C\)	1, 1	2, 2	3, 1
3-5

• In the game shown above, which of the following are the Nash Equilibria? (You will need to find best responses for both players.)
  1. (A, X)
  2. (A, Y)
  3. (B, X)
  4. (C, Z)
  5. More than one of the above.

Best Responses and Non-Rationalizability

• Finding best responses first makes it a lot easier to search for non-rationalizable strategies.
• Recall that a non-rationalizable strategy can also be called a non-best response: if none of the payoffs of a strategy are marked to indicate that it is a best-response, it is non-rationalizable. In other words:
  • Any row in which none of Player 1’s payoffs are marked to indicate a best response, is non-rationalizable.
  • Any column in which none of Player 2’s payoffs are marked to indicate a best response, is non-rationalizable.

Example: Non-Rationalizability from Best Responses

• Finding best responses first makes it a lot easier to search for non-rationalizable strategies.

• Recall that a non-rationalizable strategy can also be called a non-best response: if none of the payoffs of a strategy are marked to indicate that it is a best-response, it is non-rationalizable. In other words:

  • Any row in which none of Player 1’s payoffs are marked to indicate a best response, is non-rationalizable.

    • Any column in which none of Player 2’s payoffs are marked to indicate a best response, is non-rationalizable.

Example: Non-Rationalizability from Best Responses

		P\(_2\)
		\(a\)	\(b\)	\(c\)	\(d\)
3-6 *P\(_1\)	\(A\)	1, 0	2, 1	3, 1	4, 2
3-6	\(B\)	3, 1	2, 2	2, 0	3, 0
3-6	\(C\)	3, 1	4, 0	1, 0	2, 0
3-6	\(D\)	4, 2	3, 0	2, 1	1, 1
3-6

• We found these best responses earlier (as well as the NEs of this game): note that in the row for Player 1’s strategy B, none of Player 1’s strategies are marked.

• Likewise, none of Player 2’s payoffs are marked in column c. B and c are non-rationalizable strategies.

Example: Non-Rationalizability from Best Responses

		P\(_2\)
		\(a\)	\(b\)	\(c\)	\(d\)
3-6 *P\(_1\)	\(A\)	1, 0	2, 1	3, 1	4, 2
3-6	\(B\)	3, 1	2, 2	2, 0	3, 0
3-6	\(C\)	3, 1	4, 0	1, 0	2, 0
3-6	\(D\)	4, 2	3, 0	2, 1	1, 1
3-6

• Performing IENBR reveals additional non-rationalizable strategies:
  • Once we eliminate B, b becomes non-rationalizable.
  • Once we eliminate b, C becomes non-rationalizable.