# Applied Game Theory Lecture 5

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Applied Game Theory Lecture 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

“Cash in a Hat” game (2) Payoffs: Player 1: \$0  \$0
\$1  if match net profit \$1, -\$1 if not \$3  if match net profit \$3, -\$3 if not Match \$1  Net profit \$1.5 Match \$3  Net profit \$2 Take the cash  \$ in the hat

“Cash in a Hat” game (3) 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
2 (0,0) \$0 (1, 1.5) \$1 2 \$1 1 - \$1 (-1, 1) \$3 \$3 (3, 2) 2 - \$3 (-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)
2 (0,0) \$0 (1, 1.5) \$1 2 \$1 1 - \$1 (-1, 1) \$3 \$3 (3, 2) 2 - \$3 (-3, 3)

Backward Induction in practice (2)
(0,0) \$0 2 \$1 1 (1, 1.5) \$3 (-3, 3) 2

Backward Induction in practice (3)
2 (0,0) \$0 (1, 1.5) \$1 2 \$1 1 - \$1 (-1, 1) \$3 \$3 (3, 2) 2 - \$3 (-3, 3) Player 1 chooses to invest \$1, Player 2 matches

What is the problem in the outcome of this game?
2 (0,0) \$0 (1, 1.5) \$1 2 \$1 1 - \$1 (-1, 1) \$3 \$3 (3, 2) 2 - \$3 (-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 2 (0,0) \$0 (1, 1.5) \$1 2 \$1 1 - \$1 (-1, 1) \$3 \$3 (1.9, 3.1) 2 - \$3 (-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 be bigger than a larger share of a smaller pie

Incentive Design (2) 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

Incentive Design (3) 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)

Incentive Design (4) 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 2 (0,0) \$0 (1, 1.5) \$1 2 \$1 1 - \$1 (-1, 1 - HOUSE) \$3 \$3 (3,2) 2 - \$3 (-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”

Norman Army vs. Saxon Army Game
(0,0) fight N run fight S (1,2) N run invade fight (2,1) N run (1,2) Let’s analyze the game with Backward Induction

Norman Army vs. Saxon Army Game
(0,0) fight N run fight S (1,2) N run invade fight (2,1) N run (1,2)

Norman Army vs. Saxon Army Game
fight (1,2) S N run invade N (2,1)

Norman Army vs. Saxon Army Game
(0,0) fight N run fight S (1,2) N run invade fight (2,1) N run (1,2) Backward Induction tells us: Saxons will fight Normans will run away What did William the Conqueror did?

Norman Army vs. Saxon Army Game
(0,0) fight N run fight S (1,2) run fight (2,1) N Not burn boats run N (1,2) Burn boats N fight (0,0) fight S run N fight (2,1)

Norman Army vs. Saxon Army Game
run fight S (1,2) run fight (2,1) N Not burn boats N Burn boats N fight (0,0) fight S run N fight (2,1)

Norman Army vs. Saxon Army Game
(1,2) Not burn boats N Burn boats S (2,1)

Norman Army vs. Saxon Army Game
(0,0) fight N run fight S (1,2) run fight (2,1) N Not burn boats run N (1,2) Burn boats N fight (0,0) fight S run N fight (2,1)

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

From simultaneous to sequential moves settings
Revisiting economics 101

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

Cournot Competition (2)
Cost of production: c * q Simple model of constant marginal cost Prices: p = a – b (q1 + q2)

Price in the Cournot Duopoly Game
Demand curve Tells the quantity demanded for a given price p Slope: -b a q1 + q2

Cournot Competition (3)
The payoffs: firms aim to maximize profit u1(q1,q2) = p * q1 – c * q1 Profits = Revenues – Costs Game vs. maximization problem

Cournot Competition (4)
u1(q1,q2) = p * q1 – c * q1 p = a – b (q1 + q2)  u1(q1,q2) = a * q1 – b * q21 – b * q1 q2 – c * q1

Cournot Competition (5)
First order condition Second order condition

Cournot Competition (6)
First order condition Second order condition [make sure it’s a max]

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…

What is the NE of the Cournot Duopoly?
Graphically we’ve seen it, formally we have: We have found the COURNOT QUANTITY

q2 BR1 Monopoly NE BR2 q1 Perfect competition

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

Stackelberg Model (2) 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

Stackelberg Model (3) 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?

q2 q’’2 BR2 q’2 q1 q’’1 q’1

Stackelberg Model (4) 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

Stackelberg Model (5) 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?

Stackelberg Model (6) 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

Stackelberg Model (7) 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

Stackelberg Model (8) 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’2 BR2 q’’2 q’1 q’’1 q2 The increment from q’1 to q’’1
is larger than the decrement from q’2 to q’’2 q’2 BR2 q’’2 q1 q’1 q’’1

Stackelberg Model (9) 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

Stackelberg Model (10) 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

Stackelberg Model (11) Let’s focus on firm 2:
We now can take this quantity and plug it in the maximization problem for firm 1

Stackelberg Model (12) Let’s focus on firm 1:

Stackelberg Model (13) Let’s derive F.O.C. and S.O.C.

Stackelberg Model (14) This gives us:

Stackelberg Model (15) All this math to verify our initial intuition!

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 here Games being simultaneous or sequential is not really about timing it is about information Sometimes, more information can hurt! Sometimes, more options can hurt!

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 wins 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?

The Zermelo Theorem (2) 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 Zermelo Theorem (3) 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)

The Zermelo Theorem (4) 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 W1 L1 T T 1 1 1 L1 W1 L1 L1 L1 L1 L1 T 1 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) Example of a more complex game
What is the maximum length of the game? 1 2 2 1 1 1 2 1

Theorem proof (5) 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 1 2 2 1 1 1 2 1

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! W1 1 L1

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 definitions
Sequential move games, and their interpretation Some formal definitions

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] u (2,4) 1 2 l d 1 U (3,1) r (0,2) D (1,0) 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! u (2,4) 1 2 l d 1 U (3,1) r (0,2) D (1,0)

Example (3) Backward Induction BI :: {[D,d],r} Start from the end
“d”  higher payoff Summarize game “r”  higher payoff “D”  higher payoff u (2,4) 1 2 l d 1 U (3,1) r (0,2) D (1,0) BI :: {[D,d],r}

From the extensive form
Example (4) l r u (2,4) U u 2,4 0,2 3,1 1,0 1 2 l U d d 1 U (3,1) D u r (0,2) D d D (1,0) From the extensive form To the normal form

Example (4) Backward Induction Nash Equilibrium {[D, d],r} {[D, d],r}
(2,4) U u 2,4 0,2 3,1 1,0 1 2 l U d d 1 U (3,1) D u r (0,2) D d D (1,0) Backward Induction {[D, d],r} Nash Equilibrium {[D, d],r} {[D, u],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 (-1,0) MS F SU IN NF (1,1) OUT (0,3)

A Market Game (3) Backward Induction Nash Equilibrium (IN, NF)
(-1,0) MS F SU IN IN -1,0 1,1 0,3 NF (1,1) OUT OUT (0,3) Backward Induction (IN, NF) Nash Equilibrium (IN, NF) (OUT, F) This is a NE, but relies on an incredible threat