Applied Game Theory Lecture 5 Pietro Michiardi. Cash in a Hat game (1) Two players, 1 and 2 Player 1...

transcript

Applied Game TheoryLecture 5

Pietro Michiardi

“Cash in a Hat” game (1)

• Two players, 1 and 2• Player 1 strategies: put $0, $1 or $3 in a hat

• Then, the hat is passed to player 2

• Player 2 strategies: either “match” (i.e., add the same amount of money in the hat) or take the cash

Payoffs:

• Player 1:

• Player 2:

$0 $0$1 if match net profit $1, -$1 if not$3 if match net profit $3, -$3 if not

Match $1 Net profit $1.5Match $3 Net profit $2Take the cash $ in the hat

• Let’s play this game in class• What would you do?

• How would you analyze this game?

• This game is a toy version of a more important game, involving a lender and a borrower

Lender & Borrower game

• Let’s make a couple of motivating examples– Lenders: Banks, VC Firms, …– Borrowers: you guys having a cool project idea to

develop• The lender has to decide how much money to

invest in the project• After the money has been invested, the borrower

could– Go forward with the project and work hard– Shirk, and run to Mexico with the money

Simultaneous vs. Sequential Moves

• Question: what is different about this game with regards to all the games we’ve played so far?

• This is a sequential move game– What really makes this game a sequential move game?– It is not the fact that player 2 chooses after player 1, so

time is not the really key idea here– The key idea is that player 2 can observe player 1’s

choice before having to make his or her choice– Notice: player 1 knows that this is going to be the case!

Analyzing sequential moves games

• A useful representation of such games is game trees also known as the extensive form

• For normal form games we used matrices, here we’ll focus on trees– Each internal node of the tree will represent the

ability of a player to make choices at a certain stage, and they are called decision nodes

– Leafs of the tree are called end nodes and represent payoffs to both players

“Cash in a hat” representation

(1, 1.5)

(-1, 1)

(3, 2)

(-3, 3)

What do we do to analyze such game?

Analyzing sequential moves games

• The idea is: players that move early on in the game should put themselves in the shoes of other players

• Here this reasoning takes the form of anticipation

• Basically, look towards the end of the tree and work back your way along the tree to the root

Backward Induction

• Start with the last player and chose the strategies yielding higher payoff

• This simplifies the tree• Continue with the before-last player and do

the same thing• Repeat until you get to the root

• This is a fundamental concept in game theory

Backward Induction in practice (1)

(1, 1.5)

(-1, 1)

(3, 2)

(-3, 3)

(1, 1.5)

(-3, 3)

(1, 1.5)

(-1, 1)

(3, 2)

(-3, 3)

Player 1 chooses to invest $1, Player 2 matches

What is the problem in the outcome of this game?

(1, 1.5)

(-1, 1)

(3, 2)

(-3, 3)

Very similar to what we learned with the Prisoners’ Dilemma

The problem with the “lenders and borrowers” game

• It is not a disaster:– The lender doubled her money– The borrower was able to go ahead with a small scale project

and make some money

• But, we would have liked to end up in another branch:– Larger project funded with $3 and an outcome better for both

the lender and the borrower

• What does prevent us from getting to this latter good outcome?

Moral Hazard

• One player (the borrower) has incentives to do things that are not in the interests of the other player (the lender)– By giving a too big loan, the incentives for the

borrower will be such that they will not be aligned with the incentives on the lender

– Notice that moral hazard has also disadvantages for the borrower

Moral Hazard: an example

• Insurance companies offers “full-risk” policies• People subscribing for this policies may have

no incentives to take care!

• In practice, insurance companies force me to bear some deductible costs (“franchise”)

How can we solve the Moral Hazard problem?

• We’ve already seen one way of solving the problem keep your project small

• Are there any other ways?

Introduce laws

• Similarly to what we discussed for the PD• Today we have such laws: bankruptcy laws• But, there are limits to the degree to which

borrowers can be punished

• The lender can say: I can’t repay, I’m bankrupt• And he/she’s more or less allowed to have a

fresh start

Limits/restrictions on money

• Another way could be to asking the borrowers a concrete plan (business plan) on how he/she will spend the money

• This boils down to changing the order of play!

• But, what’s the problem here?• Lack of flexibility, which is the motivation to be an

entrepreneur in the first place!• Problem of timing: it is sometimes hard to predict up-

front all the expenses of a project

Break the loan up

• Let the loan come in small installments• If a borrower does well on the first

installment, the lender will give a bigger installment next time

• It is similar to taking this one-shot game and turn it into a repeated game– Do you recall what happens to the PD game with

repeated interactions?

Change contract to avoid shirk

• The borrower could re-design the payoffs of the game in case the project is successful

(1, 1.5)

(-1, 1)

(1.9, 3.1)

(-3, 3)

Incentive Design (1)

• Incentives have to be designed when defining the game in order to achieve goals

• Notice that in the last example, the lender is not getting a 100% their money back, but they end up doing better than what they did with a smaller loan

Sometimes a smaller share of a larger pie can bebigger than a larger share of a smaller pie

• In the example we saw, even if $1.9 is larger than $1 in absolute terms, we could look at a different metric to judge a lenders’ actions

• Return on Investment (ROI)– For example, as an investment banker, you could

also just decide to invest in 3 small projects and get 100% ROI

• So should an investment bank care more about absolute payoffs or ROI?

• It depends! On what?

• There are two things to worry about:– The funds supply– The demand for your cash (the project supply)

• If there are few projects you may want to maximize the absolute payoff

• If there are infinite projects you may want to maximize your ROI

Examples of incentives

• Incentives in contracts for CEOs– Bad interpretation, they screw up the world– Mild interpretation, they align CEOs actions towards

the interests of the shareholders• Manager of sport teams• In the middle age, piece rates / share cropping

• Incentive design is a topic per-se, we won’t go into the details in this lecture

Beyond incentives…

• Can we do any other things rather than providing incentives?

• Ever heard of “collateral”?– Example: subtract house from run away payoffs Lowers the payoffs to borrower at some tree

points, yet makes the borrower better off!

Collateral example

• The borrower could re-design the payoffs of the game in case the project is successful

(1, 1.5)

(-1, 1 - HOUSE)

(-3, 3 - HOUSE)

Collaterals

• They do hurt a player enough to change his/her behavior

Lowering the payoffs at certain points of the game, does not mean that a player will be worse off!!

• Collaterals are part of a larger branch called commitment strategies– Next, an example of commitment strategies

Norman Army vs. Saxon Army Game

• Back in 1066, William the Conqueror lead an invasion from Normandy on the Sussex beaches

• We’re talking about military strategy• So basically we have two players (the armies)

and the strategies available to the players are whether to “fight” or “run”

invade

Let’s analyze the game withBackward Induction

invade

Backward Induction tells us:• Saxons will fight• Normans will run away

What did William theConqueror did?

Not burnboats

Burn boats

run fight

Not burnboats

Burn boats

S(1,2)

Not burnboats

Burn boats

Not burnboats

Burn boats

Lesson learned

• Sometimes, getting rid of choices can make me better off!

• Commitment:– Fewer options change the behavior of others– Do you remember another setting we’ve seen in class in

which this applied?

• The other players must know about your commitments– Example: Dr. Strangelove movie

REVISITING ECONOMICS 101From simultaneous to sequential moves settings

Cournot Competition (1)

• The players: 2 Firms, e.g. Coke and Pepsi

• Strategies: quantities players produce of identical products: qi, q-i – Products are perfect substitutes

• Cost of production: c * q– Simple model of constant marginal cost

• Prices: p = a – b (q1 + q2)

q1 + q2

Price in the Cournot Duopoly Game

Slope: -b

Demand curve

Tells the quantitydemanded for a given price

• The payoffs: firms aim to maximize profit

u1(q1,q2) = p * q1 – c * q1

• Profits = Revenues – Costs• Game vs. maximization problem

u1(q1,q2) = p * q1 – c * q1

p = a – b (q1 + q2)

u1(q1,q2) = a * q1 – b * q21 – b * q1 q2 – c * q1

qqu• First order condition

• Second order condition

cbqbqa• First order condition

• Second order condition[make sure it’s a max]

22)(ˆ

caqBRq

When BR for Firm 1 is q1 = 0 ?

• We simply take the BR expression and set it to zero

• That was the perfect competition quantity…

caqBRq

What is the NE of the Cournot Duopoly?

• Graphically we’ve seen it, formally we have:

• We have found the COURNOT QUANTITY

qqqBRqBR

*11221

NEMonopoly

Perfectcompetition

caqCournot 3

Stackelberg Model (1)

• We are going to assume that one firm gets to move first and the other moves after– That is one firm gets to set the quantity first

• Assuming we’re in the world of competition, is it an advantage to move first?– Or maybe it is better to wait and see what the

other firm is doing and then react?

• We are going to use backward induction

• Unfortunately we won’t be able to draw trees, as the game is too complex

• First we’ll go for an intuitive explanation of what happens, then we’ll figure out the math

• Let’s assume firm 1 moves first• Firm 2 is going to observe firm 1’s choice and

then move• How would you go about it?

q’1q’’1

q’’2

• By definition of Best Response, we know what’s the profit maximizing strategy of firm 2, given an output quantity produced by firm 1

• Alright, now we know what firm 2 will do, what’s interesting is to look at what firm 1 will come up with

• What quantity should firm 1 produce, knowing that firm 2 will respond using the BR?– This is a constrained optimization problem

• One legitimate question would be: should firm 1 produce more or less than the quantity she produced when the moves were simultaneous?– In particular, should firm 1 produce more or less

than the Cournot quantity?

• Question: should firm 1 produce more than

• Remember, we are in a strategic substitutes setting– The more firm 1 produces, the less firm 2 will

produce and vice-versa• Firm 1 producing more firm 1 is happy

• If q1 increases, then q2 will decrease (as suggested by the BR curve)

• What happens to firm 1’s profits?– They go up, for otherwise firm 1 wouldn’t have set

higher production quantities• What happens to firm 2’s profits?– The answer is not immediate

• What happened to the total output in the market?– Even here the answer is not immediate

• What happened to the total output in the market?– Consumers would like the total output to go up,

for that would mean that prices would go down!• My claim is that the total output went indeed

up– This is a direct consequence of the BR curve

q’’1q’1

q’’2

The increment from q’1 to q’’1is larger than the decrement from q’2 to q’’2

• So, what happens to firm 2’s profits?• q1 went up, q2 went down• q1+q2 went up prices went down• Firm 2’s costs are the same

Firm 2’s profit went down

• Let’s have a nerdy look at the problem:

• Let’s apply the Backward Induction principle– First, solve the maximization problem for firm 2,

taking q1 as given– Then, focus on firm 1

p = a −b(q1 +q2)

profit i = pqi − cqi

• Let’s focus on firm 2:

• We now can take this quantity and plug it in the maximization problem for firm 1

cqqbqbqaq

• Let’s focus on firm 1:

a −bq1 −bq2( )q1 − cq1[ ] =

a −bq1 −ba − c

2b−q1

⎝ ⎜

⎠ ⎟

⎝ ⎜

⎠ ⎟− c

⎣ ⎢

⎦ ⎥q1 =

a − c

2−bq1

⎡ ⎣ ⎢

⎤ ⎦ ⎥q1 = max

a − c

2q1 −b

⎣ ⎢

⎦ ⎥

• Let’s derive F.O.C. and S.O.C.

• This gives us:

CournotNEW

• All this math to verify our initial intuition!

CournotNEW

cournotb

caqq NEWNEW

Observations (1)

• Is what we’ve looked at really a sequential game?

• Despite we said firm 1 was going to move first, there’s no reason to assume she’s really going to do so!

• What do we miss?

Observations (2)

• We need a commitment• In this example, sunk cost could help in

believing firm 1 will actually play first

Assume firm 1 was going to invest a lot of money in building a plant to support a large production: this would be a credible commitment!

Observations (3)

• Let’s make an example: assume the two firms are NBC and Murdoch trying to gain market shares for newspapers production in a city

• Suppose there’s a board meeting where the strategy of the firms are decided

• What could Murdoch do to deviate from Cournot?

Observations (4)

• An example would be to be somehow “dishonest” and hire a spy to gain more information on NBC’s strategy!

• To make the scenario even more intriguing, let’s assume NBC knows that there’s a spy in the board room– What should NBC do?

Simultaneous vs. Sequential

• There are some key ideas involved here1. Games being simultaneous or sequential is

not really about timing it is about information

2. Sometimes, more information can hurt!3. Sometimes, more options can hurt!

First mover advantage

• Advocated by many “economics books”• Is being the first mover always good?– Yes, sometimes: as in the Stackelberg model– Not always, as in the Rock, Paper, Scissors game– Sometimes neither being the first nor the second

is good, as in the “I split you choose” game

The NIM game

• We have two players• There are two piles of stones, A and B• Each player, in turn, decides to delete some

stones from whatever pile• The player that remains with the last stone

• Let’s play the game

The NIM game (2)

• If piles are equal second mover advantage– You want to be player 2– Correct tactic: You want to make piles unequal

• If piles are unequal first mover advantage– You want to be player 1– Correct tactic: You want to make piles equal

• You’ll know who will win the game from the initial setup

• You can solve through backward induction

The Zermelo Theorem (1)

• Let’s try to draw a grander lesson out of the games we’ve seen so far

• Would it be possible to state, when and if a game has a solution? In this case, would it be possible to state whether there is any advantage for players moving first or second?

• Consider a general 2 Player game• We assume perfect information– Players know where they are in the game tree and

how they got there• We assume a finite game, i.e. a game-tree

with a finite number of nodes• There can be three or fewer outcomes:

W1 (player 1 wins), L1 (player 2 wins), T (tie)

• The result (or solution) of this game is:

– Either player 1 can force a win (over player 2)– Or player 1 can force a tie– Or player 2 can force a loss (on player 1)

• This theorem appears to be trivial:– Three possible outcomes– Games are subdivided in three categories:• Those in which, whatever player 2 does, player 1 can

win (provided he/she plays well)• Those in which player 1 can always force a draw/tie• Those in which, player 1 is toast, and can only loose

Examples of games• NIM, that we played earlier• Tic-tac-toe:

– If players play correctly, you can always force a tie– If players make wrong moves, they can loose

• Checkers has a solution!– Two players– Perfect information– Finite– Three outcomes

• Chess has a solution!

• In fact, the theorem doesn’t tell you how to play, it just tells you there is a solution!

Theorem proof (1)

• We’re going to prove the theorem, in a sketchy way, as this is relates to backward induction

• Proof methodology: Induction on maximum length of a game N– We’ll start with an induction hypothesis– And we’ll prove this is true for longer games

Theorem proof (2)

• If N = 1

Theorem proof (3)

• Induction hypothesis:Suppose the claim is true for all games of length ≤ N

• We claim, therefore it will be true for games of length N+1

• Let’s take an example

Theorem proof (4)

2• Example of a more

complex game

• What is the maximum length of the game?

Theorem proof (5)

2• We have two sub-

games• The upper sub-

game: follows “1” and it has length 3

• The lower sub-game: follows “1” and has length 2

Theorem proof (6)

• By induction hypothesis (for N=3), upper sub-game has a solution, say “W1”

• Again, by induction hypothesis (N=2), lower sub-game has a solution, say “L1”

• This game has a solution, it is a game of length 1 we know already!

A more complex example

• Suppose we have an array of stones, and two players

• Sequential moves, each player can delete some stones– Select one, delete all stones

that lie above and right• The looser is the person who

ends up removing the last rock

A more complex example

• According to Zermelo’s Theorem, this game has a solution and the advantage depends on NxM, the size of the array

• Think hard about it, could come at the exam…

SOME FORMAL DEFINITIONSSequential move games, and their interpretation

Definition: Perfect Information

A game of perfect information is one in which at each node of the game tree, the player whose turn is to move knows which node she is at and how she got there

Definition: Pure Strategy

A pure strategy for player i in a game of perfect information is a complete plan of actions: it specifies which action i will take at each of its decision nodes

Example (1)

• Strategies– Player 2:

[l], [r]– Player 1:

[U,u], [U,d][D, u], [D,d]

(3,1)U

Hey, they look redundant!!

Example (2)

• Note: – In this game it

appears that player 2 may never have the possibility to play her strategies

– This is also true for player 1!

(3,1)U

Example (3)

• Backward Induction– Start from the end• “d” higher payoff

– Summarize game• “r” higher payoff

– Summarize game• “D” higher payoff

(3,1)U

• BI :: {[D,d],r}

Example (4)

2,4 0,2

3,1 0,2

1,0 1,0

1,0 1,0(1,0)

(3,1)U

From the extensive formTo the normal form

Example (4)

2,4 0,2

3,1 0,2

1,0 1,0

1,0 1,0(1,0)

(3,1)U

Nash Equilibrium

{[D, d],r}{[D, u],r}

Backward Induction

{[D, d],r}

A Market Game (1)

• Assume there are two players– An incumbent monopolist (MicroSoft, MS) of O.S.– A young start-up company (SU) with a new O.S.

• The strategies available to SU are:Enter the market (IN) or stay out (OUT)

• The strategies available to MS are:Lower prices and do marketing (FIGHT) or stay put (NOT FIGHT)

A Market Game (2)

• What should you do?

• Analyze the game with BI• Analyze the normal form

equivalent and find NE(0,3)

(1,1)IN

(-1,0)

A Market Game (3)

(1,1)IN

(-1,0)-1,0 1,1

0,3 0,3

Nash Equilibrium

(IN, NF)(OUT, F)

Backward Induction

(IN, NF)This is a NE, but relieson an incredible threat

Applied Game Theory Lecture 5 Pietro Michiardi. Cash in a Hat game (1) Two players, 1 and 2 Player 1...

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