# Econ 4011 – Game Theory

Defining characteristic of game theory
Every individual’s choices affect the outcomes of others

Why is outcome of prisoner’s dilemma odd?
– Both players choose to confess when both players could be better off if both remain silent

– However if Player 1 is silent, Player 2 knows if he confesses his payoff is much higher

– Players cannot commit to staying silent, which leads to bad outcomes

Result that it is difficult to get individuals to act outside their self-interest when others cannot commit to do the same

(4) Elements of Game Theory
(1) Players: i ∈ (1,2,….n)

(2) Actions: a ∈ A

(3) Strategies: s ∈ S

(4) Outcomes: ui(si, s_i) — note these are subscripts

Elements of game theory

(1) Players
(2) Actions
(3) Strategies
(4) Outcomes

(1) n-players in a finite game. We normally care about player i

(2) Choice set A – player will choose from this action choice set
choices are indexed by j = 1,2,….m
we normally care about generic action j

(3) Strategy is a plan to make a choice
S = strategy space
sj ∈ S = a player’s particular strategy
Strategies are indexed by j = 1, 2, ….m

Strategies cannot depend on what the other player is doing (b/c we do not know for sure what other player is doing ex ante in a simultaneous move game) but strategies can depend on an agent’s belief about what the other player will do

(4) Every set of strategies for player i is associated with some payoff

ui(si, s_i) is the payoff for player i

We denote si as the strategy of player i and
s_i as the strategies of every other player

(1) Pure strategies

(2) Mixed strategies

(1) Agent takes an action for sure – simplest strategy is for player i to choose action j for sure

(2) Mixed strategy involves randomizing your choice for some percentage of the time, generally based on the likelihood of outcomes

Strategy profile
The strategies of all players other than player i

s_i — strategy profile faced by player i

Each player cannot directly affect the strategies of others – but player i’s belief about strategy profile will affect his decision

Best response correspondence
Represents best possible strategy for player i given their beliefs of a particular strategy profile

BRi (s_i) = {argmax si} ui(si, s_i)

Given the strategies of all other players, find the strategy that results in the best utility for player i

Dominant strategy si*
Dominant strategy si* for player i is one such that it is the best response to any strategy profile the player faces

∀ s_i: BRi(s_i) = si*

No matter what my opponent is doing it is best for me to play si*

Equilibrium in Dominant Strategies
Simplest solution concept to normal form games:

∀i, ∀ s_i: BRi(s_i) = si*

For every player and every profile faced by that player, their best response does not depend on what every other play does

Predicted outcome is that everyone plays their best strategy: si*

Do all games have equilibrium in dominant strategies?
No – very specialized

Iterated Elimination of Dominated Strategies
Rational players will not play strictly dominated strategies – thus we can eliminate these strategies (i.e. the column or row for the dominated strategy for all players) to arrive at a new game without strictly dominated strategies

Nash Equilibrium
A set of strategies (s₁*,s₂*….) is NE of a normal form game if

∀i: BRi(s_i) = si*

Every player i is best responded to a given the strategy profile faced by player i

Strong condition: each player knows what the strategies everyone else will play before they are all revealed

What is the difference between Nash Equilibrium and Equilibrium in Dominated Strategies?
In NE you don’t need to be best responding to every possible strategy of other players, only the one actually being played: s_i

How to solve 2-player normal form game for pure strategy NE (in general)
(1) For every possible opponent’s strategy, find the best response to that strategy

(2) Do the same for other players

(3) If one set of strategies is the best response for both players, it’s a NE

Message of battle of the sexes NE
Even if both players aren’t indifferent across multiple outcomes, lack of commitment can lead to multiple equilibria

Both have strict preferences but can’t commit so sometimes will end off worse off

Zero sum game
When one player wins, the other loses the same amount (i.e. rock paper scissors)

Zero sum games cannot have pure strategy Nash Equilibria that have one player winning (unless one player has a strategy that always wins which does not make sense)

Self enforcement with Nash Equilibrium
Once you arrive at a Nash Equilibrium it is self-enforcing because players do not have incentives to deviate

However if you aren’t already there, can be very difficult to get there

To verify you are in a NE – verify that every player’s response to what everyone else is doing is a Best Response

Auction
Involves three parts:

(1) Set of bids

(2) Some rule assigning the object to one of the bidders

(3) Transfer of money from buyer to the seller

(2) Types of auction and (2) outcomes for payment
(1) Sealed bid

(2) Open outcry

Usually awarded to highest bidder but sometime random assingement is incorporated

Winner could either pay:

(1) First price: what they bid

(2) Second price: they pay the bid of the second highest bidder

Nash Equilibrium of Sealed Bid, First Price Auctions (perfect information)
– Individual with the highest wealth will bid the value of the individual with the second highest bidder and wins the object (due to tiebreaker)

– The individual with the second highest value bids their true value

– Everyone else bids whatever they want (they are indifferent since they will still lose)

– We can’t learn a person’s true value for having them bid for it in this type of auction because individual’s will shade their bid if they think they will be the winner by bidding lower to capture more surplus

Nash Equilibrium of Sealed Bid, Second Price Auctions

Each player has value Vn but no idea of other values

Dominant strategy equilibrium for this game is for every player to bid their true value.

Since the winner pays the bid of the second highest bidder, everyone should bid their true value and will obtain the highest value of surplus possible

Mixed strategies
Plan to take one from a set of actions with fixed probabilities

A given mixed strategy can be described by a probability of distribution over the actions

si = { A Pr(A) = Pa
B Pr(B) = Pb

i.e. action A with probability A

When will player want to mix?
A player will only mix if either

(1) they are indifferent between all their outcomes

(2) the other player is also mixing

Thus we could imagine a Nash equilibria where both players are playing mixed strategies

Mixed Strategy eqilibria and indifference
– In mixed strategy equilibria both players are indifferent across all their strategies

– Even though they are indifferent – they randomize across strategies to make their opponent indifferent

– Don’t misinterpret as optimizing behavior – they have no incentives to reach these points

– Nash Equilibria only care about whether players have some incentive to alter their action

How to find mixed strategy NE?
(1) From each player’s perspective, find mix of other strategies that would make them indifferent between their actions

(2) Resulting probabilities will be mixed strategy NE

Outline of steps to find all NE of 2-player game
(1) Eliminate all dominated strategies (both pure and mixed)

(2) Find all pure strategy Nash equilibria through the best response function

(3) Check if both players can mix to make the other player indifferent across their strategies (if they can, mixed strategy NE result)