bsp1005 lecture notes 9 - game theory i - strategic thinking in static games

1Lecture Notes 9

GAME THEORY AND STRATEGIC THINKING I:Introduction and Game Theory in Static Settings

Semester II, AY2013-2014NUS Business School

BSP1005 Managerial Economics

By Jo Seung-Gyu

Recommended Readings (later, if not now.)

- Dixit, A. and B. Nalebuff (2008), The Art of Strategy, Norton, W.W. & Company - Dixit, A. and B. Nalebuff (1991), Thinking Strategically, Norton, W.W. & Company- David M. Kreps (1990), Game Theory and Economic Modelling, Oxford Univ. Press

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Outline

Introduction, Motivating Examples and Fundamentals of a Game

Strategic Thinking in Static Games Introduction Pre-Nash Equilibrium Concepts

o Dominant Strategy Equilibriumo Iterated Dominance Equilibriumo Maximin Strategy Equilibrium

Nash Equilibrium

Application: Rational Pigs

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Game: situation in which players (participants) make strategic decisions that take into account each others actions and responses

Introduction

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Opening Question:

What is the difference between the following two problems?

I want to choose the best jazz bar to refresh tonight. I want to choose the best jazz bar while avoiding my

ex-girlfriend.

Strategic thinking is the focus of game theory.

Introduction cont.

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Game theory was developed in the early 1940s (in connection with war strategies during the cold war era) and is concerned with the formal study of rational decisions in strategic situations where conflictinterfaces with mutual dependence. Such situations are common both within firms and in market actions among firms as well as in everyday life.

price wars (e.g. collusion vs competition?) or trade wars (e.g. managed trade vs free trade)

You broke up with your girl (boy) friend a week ago. And you want her back. Should you call ahead or wait for her (his) call? What is the winning strategy? More importantly, what matters in your decision?

In these situations, players have their own objectives possibly in conflict - but also share a mutual dependence.

Motivating Examples

Example 1: A Robbery at an LA Branch of Hanmi Bank

Source: http://www.koreadaily.com/news/read.asp?art_id=855394

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(June 11, 2009)

The robbery was successful in stealing US$ 3,500 after a written threat of

Give me the money or we all die

Which part of the note was the key to the successful robbery?

Example 2: Fruit Price War in Singapore (from the reading)

At two Ang Mo Kio fruit shops, customers have been queuing to buy fruits which were going for as little as $1 for 30 apples.

After the 5-day long price war, two fruit shops lost $30,000 and $50,000, respectively.

Issues

Why is it a dilemma? How to get out of the

dilemma?

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Example 3: 2006-2009 Semiconductor Industry Chicken Game

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Brief History:

2006.10 Elpida (Japanese): First to produce 70 nano DRAM chips

2007. 01 Elpida and Powerchip (Taiwan) agree to JV

2007. 11. Elpida: announced JV with Nanya and ProMos (to beat Samsung), invested $10bln in facilities

2008. 01 DRAM prices plunge2008. 05. Taiwanese chip-makers: 13-17% loss

(started the chicken game, yet the worst losers)

2009. 12- German Qimonda reports bankruptcy

- Elpida: Reduced DRAM production- Four (Powerchip, Promos, Nanya, - - Qimonda)

under government bailout- Hynix: expanded production but eventually

negotiating rescue plan with KDB- Only Samsung maintained its market share.

2012, 01 - SK took over Hynix- Elpida reported bankruptsy- Samsung and Hynix become dominant

players, while old rivals all lost their footing.

Impact on Operating Profit

DRAM output and prices

Semiconductor Industry cont.

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Issues:

How can you be the ultimate winner?

How can the situation be seen?

FUNDAMENTALS OF A GAME

A Game Framework is composed of the following elements:

o Players: A set of rational, payoff-maximizing competitors

firms, governments or any other decision groups

o Strategies: Set of actions players choose as a game plan

o Payoffs: Value associated with each possible outcome of the game (expected) economic profits, utility or national welfare etc

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Cooperative vs Non-cooperative Games Collusion is attractive but illegal.

Therefore, we only study non-cooperative games

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FUNDAMENTALS OF A GAME cont.


Order of Decision Making and Game Forms

What matters is the information or knowledge, not the physical order of players movements .

Example: Voting Game- Voters move sequentially but is a simultaneous-moving game.

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Simultaneous Movement Sequential MovementStatic vs Dynamic Normal Form Games Extensive Form Games


Information/Knowledge Mutual Knowledge:

I know A is true and you know A is true

Common Knowledge: I know A is true. You know A is true. I (You) know (I) you know A is true. I (You) know you (I) know I (You) know . A is true.

Example Take the exam or not?

Note: Game theory distinguishes perfect information from imperfect, and complete information from Incomplete it goes beyond our level.

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Outline

Introduction, Motivating Examples and Fundamental of a Game



Nash Equilibrium


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Introduction

Static Games in Normal (or Strategic) Form

The normal or strategic forms are usually presented through a game matrix highlighting the following three key aspects of the strategic situation:

The players and their identities The feasible strategy set for each player The payoffs associated with each strategy profile

Players make decisions without knowing what others did, thussimultaneous movement game.

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Example: PRISONERS DILEMMA GAME

Two NUS students stole a book from the library.

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Are the followings clear to you in the game?

Players: Strategies Payoffs

After investigation, the district attorney gathered some information about the two suspects as partners in the crime. He holds them in separate cells and they cant communicate. Each is asked to choose after given the following game rules, which is a common knowledge.

There are four possible outcomes depicted in the matrix below. The entries are prison sentences, with As sentence first. (Plea-bargain incentive is considered.)

Outline




Nash Equilibrium


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DOMINANT SRATEGY EQUILIBRIUM

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Question: Would you confess or deny?

If a player has a strategy which gives her better payoffs than her other strategies for each of possible choice by the other player, it will be safe to assume that she will play such strategy (without the need to consider about the other players choices).

Rational choice leads the two players to both play defect (i.e. confess) even though each player's individual reward would be greater if they both played cooperatively (i.e. deny). This is why its a dilemma.

They end up doing something that hurts them both.

Prisoners dilemma game revisited:

Formalization: Concept of Dominance:

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The strategy s* is a dominant strategy if it is a players optimal choice no matter what the other players do.

His inferior strategies are called dominated strategies, which a rational player would not choose.

When every player has a strictly dominant strategy, the outcome of the game is called a dominant-strategy equilibrium.

Example:

In the prisoners dilemma game, (Confess, Confess) shares such property for both players, and thus constitutes a dominant strategy equilibrium.

COMMANDMENT

If you have a dominant strategy, use it. Expect your opponent to use her dominant strategy if she has one.

Examples of Prisoners Dilemma Game

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Pricing Game:

Real Life Example:

Fruit price war in Ang Mo Kio (Singapore): Two shops lost $30,000 and $50,000 over 5 days of price war.

You can charge high or low. All charging a high price would be the best for both, but Charge Low is a dominant strategy.

In sports: Weight-Reduction and Sudden Death

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Sudden death in weight-class sports like boxing orwrestling is not a news. And an intentional weight-reduction in an unnatural level is often a reason.

In high-school wrestling, for example, participants intentionally lose unnaturally large amounts of weight so as to compete against lighter opponents. In doing so, the participants are clearly not at their top level of physical and athletic fitness and yet often end up competing against the same opponents anyway, who have also followed this practice (mutual defection). Yet if a participant maintains their natural weight (cooperating), they will most likely compete against a stronger opponent who has lost considerable weight.

The result is a reduction in the level of competition and occasional sudden death

Analogy:

- Over-weight in Sumo?

Examples of Prisoners Dilemma Games cont.

In Human (or Animal) Behavior: The Survival of the Fittest?

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During my trip to Moscow last spring, I noticed substantially many ladies are wearing high very high heels. I asked a local lady why and the returning answer was:

After the revolution, there were fewer males than females. And there was a competition among ladies to look attractive High heel was one answer.


Pseudo-Prisoners Dilemma on TV Quiz Show: Shafted

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(Wikipedia) Shafted was a British quiz show that had a slight resemblance to The Weakest Link as it was based on answering questions and eliminating fellow contestants.

The show begins with six players. In the first round, they each must declare how much money (up to 25,000) they would like to start with. This is important as you need a large amount of money to bet on the questions during the show. The person who asked for the biggest amount of money is removed from the show before they answer a single question.

Then, the remaining five contestants answer incomplete questions that they must bet on. After all players have placed their bets, the question is completed and whoever staked the most money can answer it. If their answer is correct, then they get the money they bet added onto their score and if they answer incorrectly, then the money is taken away. After a certain number of questions (equal the number of players still in the game), the person with the most money decides who they want to remove from the game. Then after the contestant is voted out, all the contestants are given the same amount of money as the leader and the process continues until there are two contestants left.

The final round takes the form of a type of Prisoner's Dilemma. The two remaining players stand opposite each other behind podia. They are playing for the amount of money the leading player ended the main game with. Then each of the players is asked if they wish to "share" or to "shaft". This is how it works:


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Suppose the two finalists were playing for 150,000. If both players decided to shaft, then both players would walk away empty-handed. If one decided to share and the other decided to shaft, then the person who shafted would win all the money.If they both decided to share, they split the money and would each get 75,000 in this case.

Shafted cont.

. share shaft

share 50%, 50% 0%, 100%

shaft 100%, 0% 0%, 0%

The tag-line delivered by the presenter of the show is

Their fate will be in each other's hands as they decide whether to share or to shaft".

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Now, lets get live through the YouTube Video and see how two contestants play the game over 217,000!!

First half of the episode: http://www.youtube.com/watch?v=X6rwrr-opEs

Second half:http://www.youtube.com/watch?v=OmcRvtSCows

Shafted cont.

Outline

Introduction, Motivating Examples and Fundamentals of a Game



Nash Equilibrium


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ITERATED DOMINANCE EQUILIBRIUM

Dominant strategy equilibrium was about which strategy will be played for sure?

If the dominant strategy equilibrium exists, it is easy to solve. But few games have the dominant strategy equilibrium.

Then what if there is no dominant strategy equilibrium? Then, as an alternative, we can ask what strategies will not be

played? The rationale behind is clear: you wouldnt play a strategy

when there is some other strategy which always does better regardless of the rivals choices.

We call such a strategy a dominated strategy.

If dominated, you wont adopt that strategy and you would delete it out of your option. Iteratively deleting such dominated strategies can although not always give us a unique outcome called iterated dominance equilibrium.

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ITERATED DOMINANCE EQUILIBRIUM cont.

Then the following thinking process applies:

Check if a player has a dominant strategy. Check if a player has a dominated strategy. Eliminate the dominated strategies Reduce the size of the game Iterate the above procedure until you cannot proceed further

Example:

(Middle, Right) is the only outcome that survives the Iterated Dominance, thus iterated dominance equilibrium.

FirmBLeft Center Right

Top 3, 6 7, 4 10, 1FirmA Middle 5, 1 8, 2 14, 6

Bottom 6, 0 6, 2 8, 5

FirmBLeft Center Right

FirmA Middle 5, 1 8, 2 14, 6Bottom 6, 0 6, 2 8, 5

FirmBRight

FirmA Middle 14, 6Bottom 8, 5

FirmBRight

FirmA Middle 14, 6

Outline




Nash Equilibrium


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MAXIMIN STRATEGY EQUILIBRIUM

Players may sometimes play conservatively to avoid a bad outcome. When this bad outcome is a really bad one, you cannot blame them, can you?

Such risk-averse players may believe others are not necessarily rational. Then they would be willing to tradeoff between risk and return. In doing so, their objective is to choose the strategy associated with the best out of the worst outcomes.

A strategy of Player A that guarantees A the best outcome if the other player plays the strategy that is worst for A, that is, the maximum out of the minimum payoffs, is called As maximin strategy.

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MAXIMIN STRATEGY EQUILIBRIUM cont.

RIGHT is a dominant strategy for player B. Given this, player A will choose BOTTOM. (This is the only Nash equilibrium of the game, as to be seen later.)

However, what if player A doubts player Bs rationality? That is, what if player A thinks there is a slight chance for player B to choose LEFT?

- Then player A may choose Top in fear of the disastrous outcome of (-100, 5). By doing this, A can guarantee herself 8, regardless of Bs choices.

Such risk-averse attitude may force the players to choose the strategies to maximize among the guaranteed minimum payoffs. Player A will choose Top since it guarantees her the higher minimum payoff than Bottom. By the same token, Player B will choose Right. This may justify (TOP, RIGHT) as a possible game outcome, which we call maximin strategy equilibrium (outcome).

Example:Player B

Left RightTop 10, 4 8, 15

Player ABottom 100, 5 20, 10

A's Min

8

100

B's Min 4 10

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MAXIMIN STRATEGY EQUILIBRIUM cont.

Usefulness of Maximin Strategy Concept:

Given the above game structure, player B might have an incentive to fake her rationality. Sometimes, you may want to create a fog for your own good.

Skepticisms of Maximin Strategy Concept:

Very little justification for an equilibrium concept for a rational player:- The concept may sound practical, particularly when the damage from

the bad outcome is fatal. But how bad is fatal? Would you still behave conservatively when (Bottom, Left) gives (-1, 5) or (4, 5) instead of (-100, 5)?

Furthermore, It is not always self-enforcing.

In sum, this concept may be acceptable for a risk-averse behavior but inconsistent and fragile as an equilibrium concept.

Outline




Nash Equilibrium


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NASH EQUILIBRIUM

The dominance concept was about What strategies would a rational player always play or never play?

In contrast, Nash equilibrium is motivated by the question What properties must a game equilibrium have?

The Core idea suggested by John Nash was:

- No player should have an incentive to deviate from the equilibrium, once it is reached.(That is, a players strategy choice should be self-enforcing with the other players strategy choice. Each players chosen strategy should be optimal given that every other player chooses the equilibrium strategy.)

Lets look through examples below:

For the majority of games, modelers use the Nash equilibrium (named after a Princeton Mathematician and 1994 Nobel laureate John F. Nash).

This is the most important and most frequently encountered equilibrium concept, albeit it contains its own imperfectness.

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Example: Stag Hunt Game

The French philosopher Jean-Jacques Rousseau described a situation in which two individuals go out on a hunt (The Social Contract, 1762).

Each can individually choose to hunt a stag or hunt a rabbit. (It highlights the social benefit from coordinating the choices.) Hunter B

Stag Rabbit Stag 5, 5 0, 1 Hunter A Rabbit 1, 0 1, 1

Nashs suggestion is: o Put yourself in your rivals shoes. That is, choose your best response to each

of your rivals potential choices

Blue Hunter As best responseGreen Hunter Bs best response

o Nash equilibrium is a solution of mutual best response. That is, it should satisfy the players best responses at the same time.

We have Two Nash in this game: (Stag, Stag) and (Rabbit, Rabbit) Naturally, neither player has an incentive to deviate.

Note that the dominance concept does not help us.

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Finding a Nash Equilibrium: Easy Algorithm

When a game does not have any dominant or dominated strategies, or when the iterated deletion of dominated strategies does not yield a unique outcome, cell-by-cell inspection is used. Simply put, each cell of the game box is checked, each time asking "does either player have incentive to change her strategy unilaterally" or "are both players playing a best response."

The problem with this approach is that it can take quite a while. If there are two players, one with N strategies and the other with M, we need to check NxM cells which, for many games, can be quite large. For example, if each of two players has 5 strategies, we need to check 5x5=25 cells..

Below, you will learn a simpler technique for conducting cell-by-cell inspection. This method, called "strategy-by-strategy inspection" (or Dot-Algorithm, as I name it myself) requires only N+M inspections, and is less likely to result in mistakes

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Finding a Nash Equilibrium: Easy Algorithm - cont

Consider a two-player game with five strategies for each player: Player 2

F G H I JA 9 , 9 7 , 1 5 , 6 3 , 4 1 , 1B 7 , 8 5 , 2 3 , 6 1 , 4 3 , 3

Player 1 C 5 , 6 3 , 3 1 , 8 9 , 7 1 , 5D 3 , 9 1 , 9 9 , 4 7 , 9 5 , 9E 1 , 2 9 , 8 7 , 7 5 , 6 3 , 7

Use dots (or underlines or circles) to specify player 1s best response and player 2s best response. Player 2

F G H I J A 9 , 9 7 , 1 5 , 6 3 , 4 1 , 1 B 7 , 8 5 , 2 3 , 6 1 , 4 3 , 3

Player 1 C 5 , 6 3 , 3 1 , 8 9 , 7 1 , 5 D 3 , 9 1 , 9 9 , 4 7 , 9 5 , 9 E 1 , 2 9 , 8 7 , 7 5 , 6 3 , 7

You can see there are three Nash: (A, F), (E, G) and (D, J)

- Check if they are self-enforcing, i.e., if neither has an incentive to deviate.

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A Beautiful Mind and Nash Equilibrium

Now, lets see a movie clip from A Beautiful Mind to check your understanding.

Movie Clip: A Beautiful Mind

In the movie, "A Beautiful Mind," John Nash gets the idea for Nash equilibrium in a student hangout where he is sitting with three buddies. Five women walk in, four brunettes and a stunning blonde. Each of the four buddies starts forward to introduce himself to the blonde. Nash stops them, though, saying, "If we all go for the blonde, we will all be rejected and none of the brunettes will talk to us afterwards because they will be offended. So let's go for the brunettes." The next thing we see is the four buddies dancing with the four brunettes and the blonde standing alone, looking unhappy. Nash then runs to his professor to discuss his new idea of Nash equilibrium and how individual pursuit of payoff maximization does not lead to optimal results.

Lets take Russell Crowes line of If we all go for the blonde, we will all be rejected and none of the brunettes will talk to us afterwards because they will be offended as a genuine part of the game rules.

Is the result in the story a Nash equilibrium? Why or why not? If not, what would be Nash equilibrium for the situation?

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NASH EQUILIBRIUM IN MIXED STRATEGY(Its an advanced topic. We may have to skip on this section or we will just go over the concept.)

- Sports: Penalty Kicks in Soccer- Law Enforcement: DUI checking points- Olympic Games: Random drug testing

What is the common feature of the following situations?

If opponent knows what I will do, I will always lose! What would you do then?

RANDOMIZE!By randomizing just right, you can avoid getting outguessed and taken advantage of

Definition: For any given set of pure strategies, mixed strategies are rules that tell a player with what percentage to play each pure strategy.

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NASH EQUILIBRIUM IN MIXED STRATEGY cont.

Example: Consider the tennis game between Nadal and Federer.

- Nadal is serving to Federer and is deciding between down the line (DL) and cross court (CC).

- Payoffs represent the winning chance (%) for each player. Federer

DL CC DL 50, 50 80, 20 Nadal

CC 90, 10 20, 80

Notes:

- Is it clear that you will lose if you get outguessed?- Is it clear that there is no Nash equilibrium in pure strategies? - Randomizing is a solution.

Finding Nash equilibrium in mixed strategy goes beyond our level, so well skip.

Outline




Nash Equilibrium


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APPLICATION: RATIONAL PIGS

Two rational pigs, one big (called Napoleon) and the other small (called Pinkie), are put in a cage. There is a button at one end of the cage and food dispenser at the other end which dispenses 10g of food when button is pressed.

Now if the pigs want food, they need to press the button; the pig that pressed the button must run to the food dispenser while the other pig has started to eat. Napoleon, being big and dominant, can eat all the food remaining once it reaches the dispenser. So Pinkies only chance to eat is when she reaches food before Napoleon, yet still need to subordinate to Napoleon once he arrives. Pressing the button and running to the food dispenser involves calorie expenditure, which costs 2g to each pig.

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RATIONAL PIGS - cont.

If Pinkie presses the button, Napoleon eats all 10g. -------- (10, -2) If Napoleon presses the button, Pinkie can eat 7g before

Napoleon arrives. --------- (1,7) If both press the button, Pinkie can run faster (given

Napoleons fat shape) and eat 3g. ---------- (5,1) If neither presses, no food of course ---------- (0,0)

Question: Who is going to press the button?

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RATIONAL PIGS - cont.

So, who is going to press the button? Why?

Lesson:

At times, a weakness can be a strategic strength and vice versa.

- Often quoted as Judo Economics

The payoff matrix for the pigs:

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RATIONAL PIGS - cont.Business Analogy of the Pig Games: OPEC Game

The first quarter oil prices in 2013 are 10% lower than a year before.

OPEC members share a common incentive to produce less, and agree on the tight production quotas.

Saudi Arabia (SA) is a stronger player than the rest of OPEC due to its relatively large oil reserves and low production costs.

Once agreed, an individual member has a unilateral incentive to disobey the agreed quota and produce more to take advantage of the supported oil price.

If the rest of OPEC obeys, Saudi Arabia also has an incentive to cheat. But if the rest of OPEC do not obey its cartel quota (and produced too much), it is still

in the Saudi's interest to keep to its cartel quota (since SAs share is too big and SA had better support the oil price by reducing its own production)

RestofOPEC

disobeycartelquota Obeycartelquotadisobeycartelquota LOW,LOW VERYHIGH,VERYLOW

SaudiAribia

obeycartelquota MODERATE,VERYHIGH HIGH,HIGH

Highlight

SAs strength in market power may turn into the weakness in strategic power.

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Next week, well be discussing dynamic games

and strategic moves.

bsp1005 lecture notes 9 - game theory i - strategic thinking in static games

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