## Econ 4011 – Game Theory

– 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

(2) Actions: a ∈ A

(3) Strategies: s ∈ S

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

(1) Players

(2) Actions

(3) Strategies

(4) Outcomes

(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

(2) Mixed strategies

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

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

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

∀ s_i: BRi(s_i) = si*

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

∀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*

∀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

(2) Do the same for other players

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

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

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)

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

(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) 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

– 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

Each player has value Vn but no idea of other values

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

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

(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

– 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

(2) Resulting probabilities will be mixed strategy NE

(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)

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