i = \text{Player }i
-i = \text{Player(s) other than }i
\text{Example 1: Consider a strategy profile for four players }s = (s_1, s_2, s_3, s_4)
\text{If we consider player 2, then }s_i = s_2 \text{ and } s_{-i}=(s_1, s_3, s_4)
\text{Example 2: }BR_i(\theta_{-i})\text{ is read:}
\text{“Player $i$'s best response to their beliefs about other players' strategies"}
u_i(s_i,s_{-i}) = \text{Player $i$'s utility from playing $s_i$ when others play $s_{-i}$}

Before we begin: Notation alert!

Dominance and Rationalizability

Today's Agenda

Dominant and Dominated Strategies

Best Responses

Rationalizability and
Iterated Deletion of Dominated Strategies

Dominance

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.

The Prisoners' Roommates' Dilemma

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.

Best Response and Rationalizability

  • 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

Best Responses

Informally: a thought exercise that says:
"If I believe the other player(s) are going to play some way, what should I do?"

Formally:

\theta_2 = (\frac{1}{3}, \frac{1}{2}, \frac{1}{6})
u_1(U, \theta_2) =
u_1(M, \theta_2) =
u_1(D, \theta_2) =
BR_1(\theta_2) =
\theta_2 = (\frac{1}{2}, \frac{1}{4}, \frac{1}{4})

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
?

  • A strategy is rationalizable if
    it is a best response for some belief about what the other player might do.
  • Since there is no belief about what Player 2 might do for which M is a best response,
    M is not a rationalizable strategy.
  • Therefore, the set of rationalizable strategies is {U, D}.
  • Question: is there some sense in which M is a “dominated” strategy?
  • Answer: not by either pure strategy U or D, but by a mixed strategy.

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.

Iterated Deletion of Strictly Dominated Strategies

Find Your Crush at Lunch

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:

Iterated Deletion of

Dominated Strategies

  • Suppose it is common knowledge that all players are rational,
    so they will never choose a strictly dominated strategy.
  • 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?

Keynesian Beauty Contest (70% Game)

Everyone chooses an integer between 0 and 100.

The closest person to 70% of the average guess wins.

0

100

Voluntary Contribution to a Public Good

You are throwing a party.
Each person \(i\) contributes \(g_i\) anonymously.

\displaystyle{G = \sum_{i=1}^N g_i}

From player \(i\)'s perspective:

\displaystyle{G(g_i, g_{-i}) = g_i + \sum g_{-i}}

Utility depends on \(G\) and private consumption \(x_i\):

u_i(G,x_i) = G \times x_i
u_i(g_i, g_{-i}) = \left(g_i + \sum g_{-i}\right) \times (m_i - g_i)

Payoffs as functions of strategies:

Each player has income \(m_i\); private consumption is what they have after contributing.

x_i(g_i) = m_i - g_i
u_i(g_i, g_{-i}) = \left(g_i + \sum g_{-i}\right) \times (m_i - g_i)

Payoffs as functions of strategies:

Two person case: player 1 has \(m_1 = 32\), player 2 has \(m_2 = 40\).

u_1(g_1,g_2) = (g_1 + g_2)(32 - g_1) = 32g_1 + 32g_2 - g_1g_2 -g_1^2
u_2(g_1,g_2) = (g_1 + g_2)(40 - g_2) = 40g_1 + 40g_2 - g_1g_2 -g_2^2
u_1(g_1,g_2) = (g_1 + g_2)(32 - g_1) = 32g_1 + 32g_2 - g_1g_2 -g_1^2

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.

Second Strategic Tension:
Strategic Uncertainty

Conclusions and Next Steps

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.

Econ 51 Spring 2021 | 8 | Dominance and Rationalizability

By Chris Makler

Econ 51 Spring 2021 | 8 | Dominance and Rationalizability

The first steps to understanding strategic behavior: what will you NOT do? What, based on your beliefs about the other players' strategies, might you do?

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