Week 6 Flashcards
So far what are the 2 abstractions we have looked at?
1st abstraction: Perfectly competitive market
2nd Abstraction : Monopoly
These two are 2 opposite extreme cases on the spectrum.
Within the first abstraction of Perfectly competitive market, what does it include?
Each firm acts in isolation, No strategic interaction they are price takers, they don’t internalise the fact that there actions in the market do affect price, even in a small way.
Within the 2nd abstraction, monopoly what can we deduce from it?
There is no strategic interaction, it does internalise the fact that it its actions affect price, but as monopolist is the only firm in the market, they do not engage in strategic action with anyone.
What is the 3rd abstraction we are going to look at?
Duolopoly (what is the price the other firm is choosing), oligopoly( what are the other bidders going to do), auctions and kitty genevose( neighbours have to think whether to call or not to call)
So we know that the 3rd abstraction is about strategic interaction, what model will we use?
Game theory, which is a model of strategic interaction, and helps us understand what happens in between perfect competition and monopoly.
Remember in the kitty genevose game, what was the solution concept?
Rationality (Each neighbor has a belief about the behaviour of other neighbours and “best responds” to that belief.)
Beliefs are correct . (All neighbors actually behave inline with the others beliefs about them.)
What is a game ?
1) Set of players
2) Specification of actions for each player
3) Payoffs ( motivates, utiltiy functions)
In the Kitty geneovese case what was the game under the specififications?
1) Sets of players ( 38 players)
2) Specification of actions for each player( call or not call)
3) Payoff ( X , X-1 ,0)
We have 2 solution concepts in game theory which are what?
Strict domiance
Nash equilbrium
What is a Nash equilbrium?
Is a concept within game theory where the optimal outcome of a game isi where there is no incentive to deviate from their inital strategy
Draw a 2 by 2 game of the kitty genevose case?
What is the Nash equilbrium of the kitty geneovese game?
There are 2 where Neighbour calls and Neighbour 2 doesnt call
and when neighbour 1 doesn’t call and Neighbour 2 calls
This is because given that Neighbour one doesn’t call, neighbour 2 can either call or not call but there is no incentive to deviate. In addition given that neighbour 2 calls, there is no incentivate for neighbour 1 to deviate, vice versa for when neighbour 1 calls and neighbour 2 doesn’t.
If there is a Nash equilbria, what must exist?
A mixed strategy equibria
Not all games have a what action?
A strictly dominate action
Lets say you and a student study together for an exam, you either prepare before the meeting or watch great british bake off.
If both prepare you get a payoff of 60
If both you don’t prepare you get a payoff of 40
If one prepares and the others don’t the payoff of one is 35 and the other is 75
Draw the 2 by 2 game?
From this payoff matrix what can we deduce
The action of not preparing strictly dominates the action of preparing as 75>>60 and 40>>35, for both players.
So what does strictly dominate mean?
-a strictly dominant strategy is that strategy that always provides greater utility to a the player, no matter what the other player’s strategy .
Will a rational player ever play a strictly dominate action?
No as they will never pick an option that will give them a worse payoff. Rational players seek to maxmise their payoff from the game. Playing a strictly dominate action will always give them a lower payoff.
As we know not to prepare strictly dominates to prepare what does this also mean?
That not prepare not prepare is a nash equilbrium, so this is what we predict rational players will do
In a nash equilbrium can there be a strictly dominate action?
No as they are not best responding if they pick a strictly dominate action.
What else can you see from this Nash equilbria?
Both of the players would of been better off preparingf as 60,60>40, 40.
What can we say about preparing preparing having a high payoff than the nash equilbria not preparing, not preparing?
Both preparing pareto dominates us not preparing( the nash equilbria), as both of us would be strictly better off preparing.
What is the difference between a perfectly competitive firm who play this game to prepare and not prepare and in a duolopoly type people game?
In a perfectly competitive firm, we put rational people together and somehow they always achieve a pareto efficient allocation, but here they achieve something that is not pareto efficient, this is because of externalities in the game. This will always arise between the two players.
So to conclude how is perfectly competitive firm different to game theory?
Players who are rational within a perfectly compeittive market always get to the efficient outcome but in the real world, this is not true.
What is the Nash equibria here?
Well low price strictly dominates high price for both players so the Nash equibrium is Low price, Low price, in a perfectly compeititve market, they would both go for high price for higher profits.
What is the Battle of the sexes game?
Another game here:
A man and a women want to get together for an evening of entertainment, but they have no means of communication,
They can either go to the ballet or fight
The man prefers to go to the fight
The women prefers going to the ballet
But they prefer to be together than be alone
Another game here:
A man and a women want to get together for an evening of entertainment, but they have no means of communication,
They can either go to the ballet or fight
The man prefers to go to the fight
The women prefers going to the ballet
But they prefer to be together than be alone.
Payoffs
Man goes to fight 2 and women goes to fight 1
Man goes to the ballet 1 and women goes to the ballet 2
If they mismatch they both get 0
Draw payoff matrix
The row player is a women and the column player is a man.
What is the nash equilbria for the battle of the sexes
Intituevely from the start you can see that there is two nash equibria which is when they are together, either watching the fight or ballet,
What is a mixed strategy nash equibria and how is this different to pure strategy nash equilbria?
A mixed strategy exists in a strategic game, when the player does not choose one definite action, but rather, chooses according to a probability distribution over a his actions(they randomise to be unpreditctable) Note: In pure strategies, the player assigns 100% probability to one plan of action.
When will there not be a mixed strategy Nash equilibrium?
“If both players have strictly dominant actions, there is no mixed strategy equilibrium!”
For the battle of the sexes calculate the mixed strategy nash equibria?
Collum player 2p + 0x(1-p) = 0 x (1-P) = 0 x P + 1 x (1-P)
2p = 1-p
3p = 1
p = 1/3 ( you find using row)
Row player
1 (x) + 0 (1-x) = 0(x) + 2(1-x)
1x = 2-2x
3x=2
x =2/3 ( you find using column)
If there is only one such equilibrium
“There is no mixed strategy equilibrium”
If there are no pure strategy equilibria or two pure strategy equilibria, continue!
You can find a mixed strategy nash equibrium
What is the matching pennies game?
You and your friend simultaneously reveal a penny
If both pennies show heads or both show tails player 2 pays player 1 $1
If one penny shows heads and the other shows tails, player one pays player 2 $1.
You and your friend simultaneously reveal a penny
If both pennies show heads or both show tails player 2 pays player 1 $1
If one penny shows heads and the other shows tails, player one pays player 2 $1.
Draw the matching pennies matrix?
Is there a Pure strategy Nash equibrium in the matching pennies game?
No there is no pure strategy Nash equibria in this finite game, thus this means there must be a mixed strategy Nash equibria.
Why does matching pennies make sense in this game?
If we play this game, we should be “unpredictable.” That is, we should randomize (or mix) between strategies so that we do not get exploited.
Calculate the mixed strategy for the matching pennies game?
Column player P(-1) + 1(1-p) = 1P + 1(1-p)
P = 1/2 ( payoff from rows)
Row player
1(x) + -1(1-x) = -1(x) + 1(1-x)
P =1/2 ( Payoffs from columns)
What can you say about the matching pennies case?
As this is a symmetric game, the payoff The row player and column player is always going to be indifferent in between playing heads or tails 1/2 of the time.
Find nash equibria for this game?
Is there is a Mixed strategy Nash equibria for this game?
If there are 2 pure strategy Nash equilbria, there must exist a mixed strategy nash equibria.
Calculate the mixed strategy nash equibrium for this game?
Column player
3×c + 1×(1-c) = 1×c+2×(1-c)
c=1/3
Row player
2×r + 1×(1-r) ( Expected payoff for Column player playing Left: With probability r she gets 2 and with probability 1-r she gets 1) = 1×r+3×(1-r)(Expected payoff for Column player playing Right: With probability r she gets 1 and with probability 1-r he gets 3)
r=2/3
What is a Duopoly Oligopoly and Monopolistic competition?
Duopoly = 2 firms competing (pespi vs coke)
Oligopoly = several firms competiting ( big 4 accounting firms)
Monopolistic competition = each firm can still affect price ( sandwich and coffee stores)
Why do monopolist compeititve firms have a bit of market power?
They have differentatied goods.
Just to give a flavour of compeitition vs collusion so we established there is a pure strategy Nash equibria of Low price and low price, but if both pespi and coke agree on high price, they can get profit, but what is the problem?
One could easily deviate the agreement and go for higher profits of 75 and go for low price, hence collusion is hard to sustain.
Is collusion legal?
It is illegal.
Lets say there is a game where women and men invest or not invest, ( male and women in a lab and they se what makes different sexes volunteer, results find that men and men and women and women volunteer and equal rates, but when women and men are together in the room, women tend to volunteer more than mean
Lets say if both invest, the payoff is 1.25
if both do not invest the payoff is 1
If one invests and one volunteer the payoff is 1.25,2 (2 is the one that doesn’t invest)
Draw matrix
Solve the mixed strategy nash equibria for this game
From the results we can see why do women invest more than men?
Women are seen as a culture focal point and everyone excepts them to volunteer, given that everyone excepts them to volunteer, they actually do with higher probability than men, but when women are put in the same group, there is no focal point , so the proposenity to volunteer is the same as men.
Lets say we have a tennis matrix
Why is a mixed strategy useful tool to predict what happens in a tennis match when players have to decide whether to hit the server for the forehand or for the backhand of their opponents?
A mixed strategy, allows for thhe sender to make the receiver keep guessing and randomise his shots, giving a better payoff to win the game.
Assume a two-by-two game in which each player has one action that strictly dominates the other. This game:
a. Might have a mixed strategy Nash equilibrium.
b. Will not have a mixed strategy Nash equilibrium.
c. Will surely have one mixed strategy Nash equilibrium.
d. Will surely have more than one Nash equilibrium.
B
A is correct
D is correct
E is correct
The two conditions are the indifference conditions for each player: For the row player to be willing to mix, she has to be indifferent between going to Tottenham or Arsenal.
● If she goes to Tottenham she will get a payoff of 3q+0(1-q)=3q ● If she goes to Arsenal she will get a payoff of 0q+1(1-q)=1-q
● So the condition is: 3q=1-q which implies that q=1/4
The two conditions are the indifference conditions for each player: For the column player to be willing to mix, she has to be indifferent between going to Tottenham or Arsenal.
● If she goes to Tottenham she will get a payoff of 1p+0(1-p)=p
● If she goes to Arsenal she will get a payoff of 0p+3(1-p)=3(1-p) ● So the condition is: p=3(1-p) which implies that p=3/4
The answe is B and D
Question 2 Choose one (or more) of the following options:
a. The mixed strategy involves the row player and the column player playing each action with equal probability.
b. The mixed strategy involves the row player playing Fight with probability 1/3 and the column player playing Fight with probability 2/3.
c. The mixed strategy involves the row player playing Fight with probability 2/3 and the column player playing Fight with probability 1/3.
d. There is no mixed strategy equilibrium for this game.
The answer is D
A and D
What does the Betrand Versus Cournot competition emphaisis?
When you have 2 firms competing, it is not necessairly that the level of competition is going to be less, if you have 3 or 4 etc. Competition can be very fierce between 2 firms or relaxed, which depends on whether the 2 firms pick Quantites or prices.
For a perfeclty competitive firm and monopoly what do they choose price or quanitiy?
Perfectly competitive firm cannot control price, so they choose quanitity and a monopoly can choose either as it doesn’t make a difference, as it leads to same results
What are the 2 models of competition in the betrand and cournot case?
Betrand; competition on prices ( between 2 or more firms)
Cournot: competition on quantities( between 2 or more firms)
Lets look at Betrand Price competition
● Two firms compete with one another by choosing prices, p1 and p2 .
● The demand is Q = 100 − p.
● Fixed MC = 1.
● But…
○ If p1 < p2 everyone buys from Firm 1. (both goods are perfect subistutes)
○ If p2 < p1 everyone buys from Firm 2.
○ If p1 = p2 half buy from Firm 1 and half from Firm 2.
Draw matrix
This shows profit for firm 1 and 2
( 100- p1)(p1-1) this means what each firm sells each unit for price of p and at cost of 1 for each unit they sell ( the difference in revenue from each sell p1-1) ( the number of units they sell is going to be determined by demand ( 100-p1)
How can we have a nash equibrium where 1 < p1
No because firm 2 is earning 0 profit, but if it were to undercut P1, change its price from p2 to something lower than P1 but strictly higher than 1, it could get a strictly positive profit.
Can we have 1
No as frim 1 can deviate and lower price just a bit and get all the profits.
Can we have an equilbirum 1 = p1
( 1 being mc)
No firm 1 can deviate by increasing its price just a bit, still smaller than p2, everyone will still go there instead of getting 0 profit, it can still get a strictly postive profit.
Can P1 = P2 = 1 (being MC) be a nash equilbrium?
Yes, each firm is earning zero profits, but there is nothing they can do.
So what is the prediction of the betrand model?
Both firms will undercut each other to the point where both of them price at MC, meaning 0 profit, so price competiiton is very brutal.
What are the general takeaways about nash equilbrium when you have more than 2 actions? for betrand case and in general.
if you want to rule out an equilbrium, find one player who wants to deviate.
Lets look at cournot compeition on quanitites
Assume that two firms, 1 and 2, are competing
- Each firm i = 1,2 chooses how much to produce qi
- Total quantity produced is Q = q1 + q2
- The price is given by market demand P(Q) = 9 – Q
- Marginal cost 0
Profit for firm is qi X p(q)
What would be best response of firm 1, if firm 2 was choosing some q2?
Firm 1 tires to maxmise revenue which is
q1 x P(Q)
= q1 x ( 9 - q1 - q2)
Now you take the first order condition
9q1 - qi^2 - q1q2
q1 (q2) = x(y) = x=q1 y = q2
Q1 in terms of Q2
9x - x^2 - xy
( find dervative of this)
9 - 2x -y = MR
MR = MC MC = 0
9-2X-Y=0
X = Y-9/2
x= Q1
Y = Q2
q1 = (q2-9/2)
Lets look at cournot compeition on quanitites
Assume that two firms, 1 and 2, are competing
- Each firm i = 1,2 chooses how much to produce qi
- Total quantity produced is Q = q1 + q2
- The price is given by market demand P(Q) = 9 – Q
- Marginal cost 0
Profit for firm is qi X p(q)
What would be best response of firm 2, if firm 1 was choosing some q1?
Firm 1 maxmises q2 x P(Q)
q2 x ( 9-q1 - q2) = 9q2- q2q1 - q2^2
Q2 in terms of q1
9y - yx-y^2
differentiate in terms of X
9 - x - 2y - MR
MR=MC
9-x-2y = 0
y = 9-x/2
x = q2
y = q1
q2 = 9-q2/2
How do we find nash equilibrum with cournot competition?
● Looking for a profile of actions, (q1 , q2 ), such that:
○ q1 is a best response to q2
○ q2 is a best response to q1
● Therefore, it should lie in the crossing of the two best response functions
What is Nash equlibrium takeaways when looking at cournot compeitition?
To compute nash equilbria it is sometimes useful to :
Draw the best response curves
To eyeball or compute where they intersect, and the intersection point is the best response is a Nash equlbria.
What is the nash equlibria in cournot competition when Firm 1s best response to firm 2 is q1 (q2) = ( 9 - q2/2 ) and firm 2’s best response to firm 1 is q2(q1) = (9-q1)/2?
in cournot competition when Firm 1s best response to firm 2 is q1 (q2) = ( 9 - q2/2 ) and firm 2’s best response to firm 1 is q2(q1) = (9-q1)/2?, HOW DO YOU THE graph, but first find out the slope?
in cournot competition when Firm 1s best response to firm 2 is q1 (q2) = ( 9 - q2/2 ) and firm 2’s best response to firm 1 is q2(q1) = (9-q1)/2?, HOW DO YOU THE graph?
As the costs are 0 the profit is the revenue which is 3 x 3 =9>0
profits in the betrand case is 0
What are the key takeaways from the betrand vs cournot competition?
the MC for cournot was 0
Problem set
There are 2 nash equilbria
Explain in words, whilst doing this why this is a nash equibria
b. Assume that n=3, can you find a pure strategy equilibrium? Explain your answer.
Draw matrix, you don’t have to?
if player 1 and 2 believe that player 3 is going to call X>x-1, therefore they will not call and there will be a nash equilbrium when Player 1 and 2 don’t call and player 3 calls
If player 1 and 3 believe that player 2 will call , they will not call and there will be a nash equibrium when Player 1 and 3 don’t call and player 3 calls
If player 2 and 3 believe that player one will call they will not call and there will be a nash equibrium when player 2 and 3 don’t call and player 1 calls.
Or
Assume the three players are named John, Jennifer and Raul. There are three pure strategy equilibria:
Eq1: Raul Calls, everyone else does not call.
Eq2: Jennifer Calls, everyone else does not call.
Eq3: John Calls, everyone else does not call.
Assume n = 4 can you find a pure strategy nash equibrium? Explain your answer?
You don;t have to draw matrix but still draw it?
If all other players believe that player 3 will call X>X-1 for all players, therefore the other players will not call and there is a nash equibrium
If all other players believe that player 2 calls X>X-1 for all other players. Therefore the other players will not call and there is a nash equibrium
If all other players believe that player 1 calls X>X-1 for all players. Therefore the other players will not call and there is a nash equibrium.
If all other players believe player 4 calls X>X-1 for all players therefore the other players will not call and there is a nash equibrium.
OR
Assume the four players are named John, Jennifer, Raul and Tinghua. There are four pure strategy equilibria:
Eq1: Raul Calls, everyone else does not call.
Eq2: Jennifer Calls, everyone else does not call.
Eq3: John Calls, everyone else does not call.
Eq4: Tinghua Calls, everyone else does not call.
d. Now assume that n is a number bigger than 4. Find a pure strategy equilibrium. Explain your answer.
Assume the n players are named 1,2,3,…,n. There are n pure strategy equilibria:
Eq1: 1 Calls, everyone else does not call.
Eq2: 2 Calls, everyone else does not call.
Eq3: 3, everyone else does not call.
….. Eqn: n Calls, everyone else does not call.
Explain in more detail.
E) In (d) you found a pure strategy Nash equilibrium of the Kitty Genovese game. What is the probability someone calls the police in that equilibrium?
In all the equilibria we just characterized in (d), one player was calling for sure. Therefore, the probability that someone calls the police in all these equilibria is one!
f. In lectures, we focused on the mixed strategy equilibrium of the game and we saw that as the number n grows large the probability someone calls the police goes down. Compare this result to what you have learned above regarding pure strategy equilibria. Discuss which of these equilibria are better predictors of situations such as the Kitty Genovese case. Can you think of situations in which the pure strategy equilibrium is a better prediction?
In the lectures, we focused on a mixed strategy equilibrium in which the probability that someone calls the policy is smaller than one. Moreover, as the number of witnesses to the crime goes up, the mixed strategy equilibrium predicts that this probability will go down!
In (b)-(d) we focused on pure strategy equilibria that yield a very different prediction about the probability that someone will call the police. In all these equilibria this probability is one, and there is no effect of increasing the number of witnesses.
So which is a better prediction to the Kitty Genovese type of story? Mixed strategy or pure
First, let us think about what is the intuition/story behind each prediction. For example, consider the pure strategy equilibrium Eq1 in (c). If we were to explain why this is a good prediction we could say something like this “everyone knows/expects that Raul will call, this is why Jennifer and John are OK with not calling, they know Raul will. But what about Raul? Raul knows that everything is on his shoulders, as he knows that no one else will call”. So the pure strategy equilibria, relying on this coordination of expectations in the group, relies on everyone expecting one particular person to be the one calling. In contrast, the mixed strategy equilibrium puts everyone on the same level. There is no focal individual that everyone knows will call, and no-one thinks that everything is on their shoulders.
It really depends on the situation, if there is a focal point then Mixed strategy is probably better for Kitty genevose as their is no focal point to help someone.
But in situations like needing a volunteer pure strategy might be better.
Think of an example where Pure strategy equibrium is a better prediction than Mixed strategy
A game of chess, as your movements are based on other players moves.
What is a strictly dominated action?
its the opposite to strictly dominant
Answer h e and b
A mixed strategy nash equibria counts as a nash equibria
There is no strictly domdinate action as there are 2 pure strategy nash equibria
