Equilibrium
Game Theory // Fall 2025
Prof. Santetti
marcio.santetti@emerson.edu
Quick recap
Quick recap
Summarizing our concepts of beliefs, rationality, and common knowledge:
People form beliefs about others’ behavior;
People best respond to their beliefs; and
These facts are common knowledge among the players.
Equilibrium
Equilibrium
Equilibrium
What does the pair of strategies {Right, Down} mean?
- It is the game’s Nash equilibrium.
Equilibrium
Defining Nash equilibrium:
A Nash equilibrium is a set of strategies—for each player—such that no player has an incentive to deviate from their strategy, given what the other player(s) is(are) doing.
- Important: Nash equilibria are always given in terms of strategies, not payoffs.
Equilibrium
Equilibrium
- No need to use dominance every time—Use best responses instead:
Equilibrium
Reminder:
- Nash equilibria are always given in terms of strategies, not payoffs.
Some iconic games
The prisoners’ dilemma
A nice parable for many different applications in Economics and Political Science.
- Two suspects in a serious crime;
- To nail the suspects for armed robbery, the police need testimony from at least one of the suspects;
- At the police station, the two suspects are put in separate rooms.
- Each suspect is offered a deal that reduces the sentence he will get if he confesses;
- The alternative is remaining silent.
The prisoners’ dilemma
Payoff structure:
- If both confess, both get 2 years in prison;
- If one confesses and the other remains silent, the first gets 5 years in prison and the other gets only 1 year;
- If both remain silent, then the two get 4 years in prison.
The prisoners’ dilemma
- What is this game’s Nash equilibrium?
The prisoners’ dilemma
The prisoners’ dilemma
One fundamental tension from the prisoners’ dilemma:
- The clash between individual and group interests.
- Strong individual incentives can lead to group losses.
“Battle of the Sexes” (1957)
- Alex and Chris are a couple, and they need to choose where to meet this evening: Opera venue or Football game.
- The choice needs to be made while each is at work, and they have no means of communicating.
- There is only one football game and only one opera venue, so the friends will meet each other if they manage to coordinate their decisions.
- Both players prefer being together over not being together, but Alex prefers Opera to Football, while Chris prefers the opposite.
“Battle of the Sexes” (1957)
“Battle of the Sexes” (1957)
Another tension:
- Rationalizability does not require players’ beliefs to be correct.
- This is known as strategic uncertainty.
Stag hunt
- Two hunters can choose whether to hunt a stag or a hare;
- A stag provides a tastier meal, while a hare is a less filling meal;
- Hunting stags is challenging and requires mutual cooperation;
- Hunting hares is less challenging can be done individually;
- Trying to hunt a stag individually will lead to frustration.
Stag hunt
Chicken
- Two players drive automobiles toward each other at top speed;
- The one who swerves to prevent a collision is the “chicken,” and the one who keeps going straight is the “winner;”
- If both go straight, there is a collision in which both cars are damaged and both players injured;
- If both swerve, they are both “chickens” and damage their reputation.
Chicken
