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### Mixed Strategies game.

Formulation

Game played just one time and all decisions simultaneous

Formulation

- A set of N-players: I = {1, 2, …, N}, index by i
Firms, consumers, government agency, etc….

- Each player i has a set of feasible strategies Si, from which he can choose on strategy si Si.
- Students choose the number of hours to study for an exam; Si =[0,100]

- Strategy Profile is a N-dimensional vector of strategies; one for each player s=(s1, s2, …, sN). The set of all strategy profiles is

Formulation (continued)

- For a player i we are often concerned about what the other N-1 players strategies are. We denote that N-1 deleted strategy profile vector by s-i = (s1, s2,…, si-1, si+1, …., sN).
The game is played by having all the players simultaneously pick their individual strategies. This set of choices results in some strategy profile sS, which we call the outcome of the game.

3) Specify payoffs. Each player has a set of preferences over these outcomes. We assume that each player’s preferences over lotteries over S can be represented by some von Neumann-Morgenstern utility function ui(s):SR

- We often adopt this notation ui(si, s-i)
- u has the property that if there are two possible strategy profiles s’ and s’’ with the probabilities q and (1 –q), the payoff, or expected payoff is
ui(qs’+(1-q)s’’) = qui(s’) + (1-q)ui(s’’)

Payoff Matrix Representation for 2-player Games

- Two food carts, A and B, each choose one of three possible locations along a beach.
- Si = {L, M, R}
- We depict the game as a table where Cart A picks a Row and Cart B picks a Column simultaneously, and the resulting cell is the outcome
- The first number in the cell is Cart A’s VNU payoff (profit), and the second number is Cart B’s Payoff (profit).
- uA(M,R) = 18
- uB(M,R) = 10

Best Reponses

- We assume that player’s are rational; each player will choose an action which maximizes her expected utility given her beliefs about what actions the other players will choose.
- A strategy si * Si for player i is a best response by i to the deleted strategy profile s-i S-i if and only if
- si Si ; ui(si *, s-i) ≥ ui(si, s-i)
- The relationship that gives i’s best response as we vary the the deleted strategy profile is i’s best response function.
BRi(s-i) = argmaxsi Siui(si, s-i)

Best Reponses Example

- Two identical firms who can grow rice at MC = 2 and FC = 0.
- They face the inverse demand function p = 102 - .5(q1 + q2), where qi is firms i’s chosen level of production (strategy)
- Firm 1’s profit is
p1(q1, q2) = (102 - .5(q1 + q2)) q1 - 2q1.

- Profit Max is where MR = MC or
102 - q1 - .5q2= 2

- Solving for q1, we get Firm 1’s best response function
q1*(q2) = 100 - .5q2.

Best Response Example

- Recall the Food Cart Location from earlier
- Let’s look at Cart A’s best response for each of Cart B’s possible strategies
- BRA(L) = M; BRA(M) =M ; BRA(R) = M
- In a Payoff Matrix formulation it’s often helpful to denote the BR by marking the payoff in the table. Here we underline the best response

Solution Concepts – How people (should?) (will?) play the game.

- Consider two strategies si’, si Si ; Strategy si’ strongly dominatessiif si’ gives player i a strictly higher expected utility than does si for every possible deleted pure-strategy profile s-i S-i which her opponents could play
- s-i S-i ; ui(si’, s-i) ≥ ui(si, s-i)

- A rational player will never play a dominated strategy
- If there exists a strategy si’ Si that strongly dominates all other
si Si\{si’ }, Then si’ is a strongly dominant strategy for i.

- A rational player who has a
dominant strategy must play it.

- Ex. Middle is a Dominant
Strategy for Cart A

Observations game.

Dominant strategy for A & B is to advertise

Do not worry about the other player

Equilibrium in dominant strategy

Firm B

10, 5

15, 0

Don’t

Advertise

Advertise

Advertise

6, 8

10, 2

Firm A

Don’t

Advertise

Payoff Matrix for Advertising GameDominance Solvable Games game.

- If every player i has a Dominant Strategy si* , then the games has a dominant strategy solution with dominant strategy profile
(s1*, s2*, …, sN* )

- All we assume is that players are rational!
- Food cart location game has a dominant strategy solution (M, M)
- Prisoner’s Dilemma Game (Defect, Defect)

Economic Applications of Dominant Solvable Games game.

- Voluntary Contribution Mechanism
- Median Voter Mechanism
- Second Price Auction or English Auction

Eliminating Dominated Strategies game.

- The other side of the coin: If we play strongly dominant strategies, then we must NOT play dominated strategies
- Now let’s build an algorithm based on this concept
- Step 1: We loop through each player and eliminate each of their dominated strategies, if no player has a dominated strategy the stop.
- Step 2: We form a reduced “new” game that consists of yet to be eliminated strategies.
- Return to Step 1:
- Once the algorithm stops the remaining set of Possible Strategy Profiles is the set of Rationalizable of outcomes.
- If the set of rationalizable outcomes consists of a single strategy profile, the game is said to be dominance solvable.

Economics Example – Bertrand Price Competition game.

- Two firms that are perfect substitutes: Imagine two toll highway toll booths next to each other.
- Let’s suppose the MC = 10 per car. And the market demand is Q = 100 – min{p1,p2} (all cars go to the cheapest toll booth(s?), if there is a tie then a car is equally likely to go to either toll booth)
- Price has to be an integer
- The monopoly price is found where MR =MC or Pm = 45
- Any price above 45 is dominated. Why?
- Then we can successively eliminate at dollar at each iteration. Why?
- The game is dominance solvable with strategy profile (11,11)

Keyne’s Beauty Contest game.

- Every student in class today is a player
- Everyone will pick a single number between zero and one hundred
- Write this down on a piece of paper with your name (and don’t show anyone)
- The TA’s will collect these slips of paper.
- We will rank the guesses from highest to lowest and then calculate the median guess
- The student whose guess is closest to 50% of the median is the winner of 50 Yuan.
- Please complete this task now and think carefully about your guess

Keyne’s Beauty Contest game.

- What strategies are dominated?
- every guess in the interval (50,100]
- Then in the restricted game where the strategy space is [0, 50], we can eliminate (25, 50]
- Repeat infinitely and the limit is the strategy profile where everyone submits zero!
What did we do?

Equilibrium Solution – Nash Equilibrium game.

- Many games don’t have a unique rationalizable outcome.
- In such cases we also need to model the player’s beliefs about what the other player’s selected strategies will be.

Definition of a Pure Strategy Definition game.

A strategy profile s* is a Nash equilibrium of a game if every player i is playing a best response against their deleted strategy profile s-i*.

- i I, si* Bri(s-i*)
- i I and for every si Si, ui(si*, s-i*) ≥ ui(si, s-i*)
So we can interpret a Nash Equilibrium as a self enforcing agreement – Given a NE s* no one player can deviate unilaterally and make themselves better off.

ui(si*, s-i*)

Nash Equilibrium Examples game.

- Payoff Matrix: Under line the best
responses and the NE correspond

to the cells with mutual best responses.

In this game there are two NE (L,L)

and (R,R). Which of these NE

play depends upon coordinating

our beliefs.

- In the cart location game
there is a unique NE at

(M,M)

- As this example suggests, any
unique rationalizable outcome

is also a NE.

Nash Equilibrium and Imperfect Competition game.

- Cournot competition, or quantity competition
- Two identical firms who can grow rice at MC = 2 and FC = 0.
- They face the inverse demand function p = 102 - .5(q1 + q2), where qi is firms i’s chosen level of production (strategy)
- Solving for qi, we get Firm i’s best response function
qi*(qj) = 100 - .5qj.

- A Nash equilibrium is a pair of production levels (q1*, q2*) that are mutual best responses to each other. Thus we need to find the solution to the simultaneous pair of equations
- q1* = 100 - .5q2*
- q2* = 100 - .5q1*
Substitution gives us

q1* = 100 - .5(100 - .5q1*) => q1* = (200/3) and by symmetry

NE is (q1*, q2* )= (200/3. 200/3)

Economics of Cournot Equilibrium game.

We can see the non-cooperative NE yields less joint profit maximazation

With a Monopoly q1*=100

As q2 increases it decrease demand

for q1

q2

P

Competitive Eq: q1+q2 =200 and profit = 0

Cournot NE: 2q*=133.33, p=33.33 and profit = 2088.66

Cooperative. 2q*=100, p= 52, and profit=2500

200

102

q1*(q2)

P(q2=0)

100

P(q2=80)

80

qNE

MR(q2=0)

q2*(q1)

MR(q2=80)

MC

60

100

120

qNE

100

200

q1

200

Q

60

Price Competition in Differentiated Goods Market game.

- Two firms competing on price, assume MC = 0 for both firms
- Symmetric differentiation the Demand curve for qi is
qi (pi,pj) = 80 – pj – 2pi.

- Notice the goods are compliments in this example and the cross price effect is negative.
- Profit for the firm is simply total revenue
ui (pi,pj) = pi(80 – pj – 2pi).

- The first order condition for profit max is
80 – pj – 4pi = 0.

Price Competition in Differentiated Goods Market game.

- So i’s best response function is
pi*(pj) = 20 – pj/4

- Since it is symmetric solving the mutual best response price is
pi*(pj) = 20 –(20-pi*/4)/4.

- Thus p*= 16 and the Nash equilibrium is (p1, p2) = (16, 16)

Coordination and Multiplicity of Nash Equilibria game.

- Many problems in economics has coordination problems.
- Consider the story of the hunter society where hunters can collect rabbits individually or collect horses cooperatively
- There is a safe Nash equilibrium (r,r) and the risky but Pareto optimal Nash equilibrium (h, h)

The Game of Tennis game.

- Server chooses to serve either left or right
- Receiver defends either left or right
- Better chance to get a good return if you defend in the area the server is serving to

Game Table game.

Game Table game.

For server: Best response to defend left is to serve right

Best response to defend right is to serve left

For receiver: Just the opposite

Nash Equilibrium game.

- Notice that there are no mutual best responses in this game.
- This means there are no Nash equilibria in pure strategies
- But games like this always have at least one Nash equilibrium
- What are we missing?

Extended Game game.

- Suppose we allow each player to choose randomizing strategies
- For example, the server might serve left half the time and right half the time.
- In general, suppose the server serves left a fraction p of the time
- What is the receiver’s best response?

Calculating Best Responses game.

- Clearly if p = 1, then the receiver should defend to the left
- If p = 0, the receiver should defend to the right.
- The expected payoff to the receiver is:
- p x ¾ + (1 – p) x ¼ if defending left
- p x ¼ + (1 – p) x ¾ if defending right

- Therefore, she should defend left if
- p x ¾ + (1 – p) x ¼ > p x ¼ + (1 – p) x ¾

When to Defend Left game.

- We said to defend left whenever:
- p x ¾ + (1 – p) x ¼ > p x ¼ + (1 – p) x ¾

- Rewriting
- p > 1 – p

- Or
- p > ½

Server’s Best Response game.

- Suppose that the receiver goes left with probability q.
- Clearly, if q = 1, the server should serve right
- If q = 0, the server should serve left.
- More generally, serve left if
- ¼ x q + ¾ x (1 – q) > ¾ x q + ¼ x (1 – q)

- Simplifying, he should serve left if
- q < ½

Mixed Strategy Equilibrium game.

- A mixed strategy equilibrium is a pair of mixed strategies that are mutual best responses
- In the tennis example, this occurred when each player chose a 50-50 mixture of left and right.

General Properties of Mixed Strategy Equilibria game.

- A player chooses his strategy so as to make his rival indifferent
- A player earns the same expected payoff for each pure strategy chosen with positive probability
- Funny property: When a player’s own payoff from a pure strategy goes up (or down), his mixture does not change

Minmax Equilibrium game.

- Tennis is a constant sum game
- In such games, the mixed strategy equilibrium is also a minmax strategy
- That is, each player plays assuming his opponent is out to mimimize his payoff (which he is)
- and therefore, the best response is to maximize this minimum.

Does Game Theory Work? game.

- Walker and Wooders (2002)
- Ten grand slam tennis finals
- Coded serves as left or right
- Determined who won each point

- Tests:
- Equal probability of winning
- Pass

- Serial independence of choices
- Fail

- Equal probability of winning

Battle of the Sexes game.

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