2.3 Game theory Flashcards
What is game theory?
Game theory is the study of strategic interaction
What is a game?
A game consists of 3 things:
• Players e.g. firms
• Strategies i.e. ‘plans of action’ (not the same as actions)
• Payoffs e.g. profits that depend on strategies of all players
Today will we look at only one shot( static) games?
No we will use look at repeated games
What 2 types of games are we going to look at and do they give the same result?
simultaneous game or sequential game, and we can get very different results.
Tell me what the players, strategies and payoffs are ?
The players are firms 1 and 2
The players’ strategies are large output or small output
The payoff for each player depends on the choice of strategy by all players.
What is a dominant strategy?
a dominant strategy is the course of action that results in the highest payoff for a player regardless of what the other player does.
What is a dominated strategy?
A strategy is dominated if there always exist a course of action which results in higher payoff no matter what the opponent does.
Is there a dominant strategy here?
Yes playing large for both players is a dominant strategy.
What is a Nash equilibrium?
The Nash Equilibrium (NE) of the game is the set of strategies that are best responses to each other.
What is the Nash equilbrium of this game ( what are the set of strategies and payoff’s?
The NE outcome: (large, large)
What payoffs do the players get in
this outcome? (16, 16)
Can both firms do better? what would they have to do to get this payoff? Is it a Nash equilibrium, is there incentive to deviate?
Yes ( 18,18) ( small,small)
This will involve them colluding to get this payoff but the problem of this payoff is that there is an incentive to deviate ( cheat), thats why collusion is inherently unstable.
What does best response mean?
A best response for player k is a strategy that maximises k’s payoff, given the strategies of the other players.
What does best response mean?
A best response for player k is a strategy that maximises k’s payoff, given the strategies of the other players.
What type of game is this?
In the Prisoner’s Dilemma, the outcome when both players act in their
individual interest is worse for both of them than if they act cooperatively.
If the 2 firms cooperate there is an incentive to cheat, hence explains why cartels are difficult to sustain.
What can we think this as a model of and what does it mean if the 2 firms collaborate (what does it do to their profits?
Think of this as a model of a cartel with two firms
• Limiting production (small, small) increases profits for all firms – firms act
jointly as a monopolist to maximise industry profits
Is there a Pure strategy Nash Equilibrium in this game and explain the strategies of each player? ( -100 is a fine, -5 is the wasted effort he exerts when he patrols and the driver parks legally, -10 is the effort of the driver to park legally, 0 is he parks illegally he doesn’t get anything if the warden doesn’t patrol.
No they isn’t a Pure strategy Nash equilibrium.
If the warden patrols the driver parks legally
If the warden does not patrol the driver parks illegally
If the driver parks legally the warden does not patrol
If the driver parks illegally the warden patrols
What is the difference between a pure strategy and a mixed strategy using this game as an example?
A player plays a pure strategy if he or she does not randomize, e.g. patrols
or does not patrol.
A player plays a mixed strategy if he or she randomizes, e.g. patrols with
probability “ 1/3 and does not patrol with probability 2/3.
Be careful if the question says to find nash equilibrium, what does that mean?
This asks you to find all the pure strategies and mixed strategies nash equilibrium.
When finding Mixed strategy Nash equilibrium, for player 1 ( row player, we calculate mixed strategies how) and what about player 2( column player
We assign probabilities to each player
1) for the row player( player 1 we use the 1st numbers in a payoff matrix of each row, using the algebraic probabilities corresponding on top of player 2)
2for the column player ( player 2 we use the 2nd number in the payoff matrix but in the downward column, using the algebraic probabilities of player 1)
Find the mixed strategy Nash equilibrium of this game and what is the calculation essentially saying?
The expected payoff from patrolling = the expected payoff of not patrolling
The expected payoff from parking legally = the E(P) from parking illegally.
If they didn’t equal it would be a pure strategy nash equilibrium.
Lets interpret this, what can we say about the reward of patrolling and finding the driver park illegally and punishment for the car driver patrolling illegally?
If the reward (15) for catching the driver park illegally goes up, the traffic warden more likely to patrol.
If the fine goes up then the traffic warden will patrol less frequently as the fine is huge, meaning drivers less likely to park illegally.
Can games have multiple equibria and what does this mean?
Yes they can and this means that game theory cannot always predict an outcome.
Find all the nash equibria for this game and what type of game is this( can we predict whats going to happen) , what does the biologist prefer and what does the economist prefer?
Two NE in pure strategies: (Mac, Mac) and (PC, PC)
• Coordination game(t describes the situation where a player will earn a higher payoff when he selects the same course of action as another player.)( we cannot predict whats going to happen)
Biologist prefers Mac and the Economists prefers the PC.
Just for reminder what is a simultaneous game?
players choose their strategies simultaneously( same time) and we used payoffs payoff matrix to illustrate the game.
Now we will look at a sequential move game ( extensive form game) what is this and give an example?
Is a way of describing a game using a game tree, it is a diagram that shows what choices are made at different points of time ( sequential, not simultaneously( same time) . This is response that siultaneous games have multiple equibria so we cannot tell what is the best outcome.
Chess is an example of a sequential game.
Is there perfect information with a sequential game ( what about simultaneous games) ?
information is perfect since each player can see the decision taken by the previous player, ( for simultaneous games, there isn’t perfect information)
We are now going to look at examples of sequential games to build. Looking at a 2 stage game.
We have 2 players, player 1 and player 2, each players will be able to choose strategies, which are strategy 1 and 2? Build the game tree and highliglight some key terminology.
1) Player 1 has to made a decision first and this is represented by a decision node ( black dot). He can pick 2 strategies ( A and B)
2) 2 new nods are drawn for player 2 ( player 2 knows the strategy of player 1, so he can pick his strategies based on player 1 picking his strategy )
3) So if player 1 picks strategy A, player 2 would have to pick strategy A or B and same thing if player 1 picks strategy B)
4) We would have payoffs at the end.
Lets look at another example of an extensive form game?
( The game has 2 stages)
Stage 1: Potential entrant chooses whether to enter
Stage 2: incumbent chooses whether to fight
Draw a game tree?
Don’t worry about the payoffs yet.( but what player does each payoff correspond to?)
Player 1 = Potential entrant
Player 2 incumbent
( The first payoff corresponds to player 1 and the second player 2)
Draw a game tree for this
Draw the tree diagram for this payoff matrix? ( why cant we use this to find sub game perfect equilibrium?
Because there is no perfect information for simultaneous games.
What are the steps to find sub game Perfect equibirum?
1) We look at each sub game using ( at the last stage. in this example its stage 2, where there are 2 subgames)
2) First sub game if you are the incumbent and you know the potential entrant has entered, you choose whether to fight or not fight, the best thing to do is not fight as 1>0. ( remember the second payoff is responds to player 2)
3) Then we look at the second sub game. we know that the potential entrant hasn’t entered so the incumbement again can choose to fight or not fight. The best thing to do here is not clear as they are indifferent. Thus we are down to 3 potential choices.
4) Now we look at the perspective of player 1 ( potential entrant), looking forward you know if you enter and the incumbent wants to fight you will not want to do that as the payoff is 0.
5) Now the optimal we look at the 3 choices we have left and we see entering not fight is optimal as he gets a payoff of 1 which is greater than >00.
Represent the subgame perfect equilibrium
and the payoffs from SPE
In the bracket you put choose of action first from player 1, then put the 3 options which were left down to we write it in a backward way)
Subgame Perfect Equilibrium: (E; NF | E, F | NE, NF | NE)
Payoffs from SPE: (1, 1)
Find the Subgame perfect equilibrium ( explain in your head)
( D; R | D, L | U)
Payoffs from SPE: (3,1)
In a simultaneous game what would be the pure strategy nash equibria as supposed to the sequential move entry game?
Simultaneous move entry game: two pure strategy Nash equilibria
(Not enter, Fight) and (Enter, Not fight).
• Sequential move entry game: one equilibrium outcome (Enter, Not fight).
What is the credible threat in this game?
T
Assuming all of this, find the monopoly profit in the market and the price they would set and quantity to make profit
1) Set out the profit equation TR - TC/ or ( P-MC/C)Q
p =10 - Q
( 10-Q-2)Q = (8-Q)Q = 8Q-Q^2
2) Firms wants to maxmise profit so we differentiate the profit function and set it 0( we also find SOC to confirm its a maximum ( if its negative its a maximum)
3) 8-2Q=0 —- Q^m = 4
P = 10-4 = 6
We now know that monopoly profit = 16 ( P-MC/C)X Q = (6-2)4 = 16.
They both set price of 6 and share profits, so each firms profit from collusion is 8
Collusion game: Now we need to think whether the firms have an incentive to deviate and cheat, so what do we need to do and solve it?
( Suppose a firm can cheat by undercutting price by £1 and gets all the market, while the other stays faithful and sets the monopoly price) find the profits of the player that cheats and the player that doesn’t?
We know that the new price = £5 that means P=10-Q, where Q = q1 + q2, 5 = 10-p =
Q = 10-5=5
Q is now the quantity of the market so q1 = Q
Profit for the firm that cheats is ( P-MC/C)XQ = (5-2)5 = 15 or use this equation 8Q-Q^2 = 40-25= 15
Profit for the firm that doesn’t cheat = 0. ( doesn’t get any of the profit because products are identical.)
If both cheat they still set price 5, but Q=5, Is divided by 5 ( quantity demanded by market.)
So ( 5-2)2.5 = 7.5
Summarise the payoffs of this collusion game?
If they both collude they will set a price of 6 a quantity ( Qd for both will be 4, each) and profits will be 8.
If one cheats and the other doesn’t, the one that cheats gets a profit of 15>8, so there is an incentive to cheat, and the other gets 0 profit.
If they both cheat then they both get profits of 7.5<8.
If they both collude they will set a price of 6 a quantity ( Qd for both will be 4, each) and profits will be 8.
If one cheats and the other doesn’t, the one that cheats gets a profit of 15>8, so there is an incentive to cheat, and the other gets 0 profit.
If they both cheat then they both get profits of 7.5<8.
Show this in a payoff matrix( example of the prisoners dilemma game and find Pure strategy Nash Equilbria?
Use backward induction to solve what the Sub Perfect Equilibrium is? ( remember all prior moves are observable) and what is the general result?
1) We know that at the final subgame, players must play a nash equilibrium in all SPE ( CHEAT, CHEAT)
2) Actions here cant affect the future cheating, you anticipate profits in the final subgame, meaning you need to be maximising now ( which is both cheating)
3) similarly at the first stage, your actions cant affect future play, so again they need to be maximising, they know they will get a flow of profits 7.5. in the future.
So generally the SPE of the finitely repeated PD is for the domiant strategy ( cheat) to be played in each stage.
So in the prisioners dilemma it seems that cheating is inevitable, but now we are going to use if the game is repeated an infinite number of times, this may not always be the case. What do we need first?
Discount factor, a HIGH DELTA MEANS VERY PATIENT ( DISCOUNTING FACTOR LOW)
A LOW DELTA MEANS VERY IMPATIENT( DISCOUNTING FACTOR IS HIGH)
Show the payoffs of firm 1 if they collude with firm 1 forever and if they cheat ? We know if they collude profit is 8 and if they cheat profits = 15.
Recall how can we turn this infinite geometric sequence into an equation ( sum of a geometric sequence)
a/1-r
where a is the first term and r is common ratio.
When is it optimal to collude?
if the present discounted value of the ‘collusion stream > that of the cheating stream. ( we don’t need to work out PV of 15 as its in the first period but watch out in the questions, there might be something in the second period, where you need to discount)
remember delta = 1/1+r so we can show the stream of cash flows, turn it into a geometric sequence, then get the sum of geometric sequence being >15.
We can rearrange to find what delta needs to be bigger than in order for collusion to be sustainable.
How can we interpret this result?
Collusion is only optimal if delta is greater than 7/15 ( ie firms are sufficiently patient, to get income streams in the future) - remember delta = 1/1+r and if r is low, delta is high)
Is delta is lower than 7/15, firm 1 will cheat ( i.e if firm 1 is not patient, to get income streams in the future) - remember delta = 1/1+r and if r is high, delta is low.
If delta = the amount of patience firm 1 has, then firm 1 is indifferent on whether they cheat or not.
Show the payoffs of profits when there are n number of firms, when they one firm colludes or cheat ( the payoffs of profits are representative) Remember monopoly profit = 16 and if you cheat you get the market so profit = 15.
We know that firms will collude if e if the present discounted value of the
‘collusion stream’ exceeds that of the ‘cheating stream’, show the geometric sequence and then rearrange to get n on one side?
What would be the number of firms collusion is sustainable with?
n<5.33 so up to 5 firms.
What is Folk theorem?
(iii) Now suppose the firms can choose not to enter the market (or else enter with a specific technology), so the game is augmented as in the matrix below. Does this game have any pure strategy Nash equilibria?
(iv) Now suppose the firms make decisions sequentially in a two-stage game. In the first stage Firm 1 decides whether to enter the industry, and, if it enters, it
chooses between a craft technology and an automated technology. In the
second stage Firm 2 chooses whether to enter the industry and, if it enters,
choose which technology to us.
This is a coordination game, they could be friends and they want to study together.
1) The first payoff in the cells is Amil. His highest payoff is 80. and we look vertically down and we notice 80>30, so the best combination for him, is when Beth and him do course 1. ( do not write the both payoffs it is wrong)
2) When we look at Beth, the second payoff in the cells is hers. Her highest payoff is 80, we look horizontally to find this is where both of Amil and her are doing Course 2.
3) Who does best if the students do different courses? This means they cant study together. If Amil does course 2 and beth does course 1, beths payoff is higher and if we look at when Amil does course 1 and beth does course 2, beth ‘s payoff is still higher. No matter what beth does better.
Use quick defintiions just as reference points.
Nash equilibrium = is a set of strategies, one for each player, that are best responses to each other.
PSNE = A strategy that a player plays with certainty
MSNE = A player who randomises their strategy.
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