Wstęp - PowerPoint Presentation

Slide1

Wstęp do Teorii Gier Slide2

Labour union vs factory management

The management of a factory is negotiating a new contract with the union representing its workers

The union demands new benefits:

One dollar per hour across-the-board raise (R)

Increased pension benefits (P)

Managements demands concessions:

Eliminate the 10:00 a.m. coffee break (C)

Automate one of the assembly checkpoints (reduction necessary) (A)

You have been called as an arbitrator.Slide3
Slide4

Eliciting preferencesManagement ordinal preferences

Further questions:

Indifferent between $0.67 raise and granting pension benefits

0.67R=P, hence P=-2 and R=-3

Willing to trade off a full raise and half of pension benefits for elimination of the coffee break

R+0.5P=-C, hence C=4

Management cardinal utility

Labor union cardinal preferences Slide5

The gameWe assume that these utilities are additive (strong assumption)

We get the following tableSlide6

Finding Nash solutionkjhSlide7

IssuesWhat if the Nash point is a mixed outcome? E.g. (2,2½)= ¼PRC+ ¾PRCA.

How to give ¾ of the automation?

Possibilities: grant automation but require that ¼ of the displaced workers be guaranteed other jobs.

What to do if there are no outcomes which are Pareto improvement over SQ?

Recommend SQ

Or better,

e

nlarge the set of possibilities – brainstorming with LU and managementIs the present situation a good SQ?Real negotiation often take place in an atmosphere of threats, with talks of strikes and lockout (each side tries to push SQ in her more favorable direction)What about false information about utilities given by each side?

E.g. correct scaling for positive and negative utilities separately, but to misrepresent the “trading off” of the alternativesSlide8

Management false utilitiesSuppose, the management misrepresents by doubling negative utilities:Slide9

The new Nash point is at (1,½) It could be implemented as:

½PC + ½RCA. In the honest utilities this point corresponds to (3½,½) -not Pareto optimal, but better for the management than (3,2)

Or ¾PC + ¼C. In the honest utilities it corresponds to (2½,½), which is worse than (3,2) for both. Slide10

Other casesAssume that now the management is truthful and Labor Union lies by doubling its negative payoffs

The solution RC (LU does not profit)

Assume that both lie and double their negative utilities

The solution SQ!!! (No profitable trade at all)Slide11

(3,2) PRCASlide12

(1,0.5)

RCA

PC

In real utilities (3.5,0.5)Slide13

(1,1) RC

In real utilities: (1,2)Slide14

(0,0) SQ

In real utilities the same (0,0)Slide15

An introduction to N-person games

Let’s consider a three person 2x2x2 zero-sum gameSlide16
Slide17

Players may want to form coalitionsSuppose Colin and Larry form a coalition against Rose

-4.4 – this is the worst Rose may get (it is her security level)

Colin should always play B and Larry 0.8A+0.2B. Slide18

Now two remaining possible coalitionsRose and Larry against Colin

Rose and Colin against LarrySlide19

Which coalition will form?How the coalition winnings will be divided?

For example in a) Colin and Larry win 4.4 in total, but the expected outcome is:

It is Larry who benefits in this coalition!

Colin though not very well off, is still better off than when facing Rose and Larry against him.

The rest of the calculations is as follows:Slide20

Which coalition will form?For each player, find that player’s preferred coalition partner.

For instance Rose would prefer Colin as she wins 2.12 with him compared to only 2.00 in coalition with Larry.

Similarly Colin’s preferred coalition partner is Larry

Larry’s preferred coalition partner is Colin.

So Larry and Colin would form a coalition!

Unfortunately, it may happen that no pair of players prefer each otherSlide21

Transferable Utility (TU) modelsVon Neumann and Morgenstern made an additional assumption: they allowed

sidepayments

between players

For example Rose could offer Colin a

sidepayment

of 0.1 to join in a coalition with her – effective payoffs (2.02,-0.59,-1.43)

This coalition is more attractive to Colin than Colin-Larry coalition

The Assumption that sidepayments are possible is very strong:It means, that utility is transferable between players.It also means, that utility is comparable btw. players.Reasonable when there is a medium of exchange such as money.Slide22

Cooperative game with TUWe assume that:

Players can communicate and form coalitions with other players, and

Players can make

sidepayments

to other players

Major questions:

Which coalitions should form?

How should a coalition which forms divide its winnings among its members?Specific strategy of how to achieve these goals is not of particular concern hereRemember going from extensive form game to normal form game, we needed to abstract away specific sequence of movesNow in going from a game in normal form to a game in characteristic function form, we abstract away specific strategiesSlide23

Characteristic function

The amount v(S) is called

value of S

and it is the

security level of S:

assume that S forms and plays against N-S (the worst possible), value of such a game is v(S)

Example: Rose, Colin and Larry

Zero-sum game since for all S: An important relation:Slide24

ExampleskjhnSlide25

ExamplesN={members of the House, members if the Senate, the President}

v

(S)=1

iff

S contains at least a majority of both the House and the Senate together with the President, or S

conatins

at least 2/3 of both the House and the Senate.

v(S)=0 otherwiseThe game is constant-sum and superadditive.Slide26

Elections 1980

Three candidates:

Democrat Jimmy Carter,

Republican Ronald Reagan,

Independent John Anderson. Slide27

PoliticsIn the summer before the election, polls:

Anderson was the first choice of 20% of the voters,

with about 35% favoring Carter and

45% favoring Reagan

Reagan perceived as much more conservative than Anderson and Anderson was more conservative than Carter.

Assumption: Reagan and Carter voters had Carter as their second choiceSlide28

If all voters voted for their favorite candidate, Reagan would win with 45% of the vote.However it may be helpful to vote for your second candidateBut, it is never optimal to vote for the worst

Suppose each voters’ block has two strategies

Three

equilibria

: RCC (C wins) and RAA, AAA (A wins)!!!

Observe that the sincere outcome RAC (R wins) in not an equilibrium.Slide29

The game may be simplified: Reagan voters have a dominant strategy of R

Sincere outcome: upper left

Carter and Anderson voters could improve by voting for their second choice

In the summer and fall of 1980 the Carter campaign urged Anderson voters to vote for Carter to keep Reagan from winningSlide30

Another example

In march 1988 House of Representatives defeated a plan to provide humanitarian aid to the US backed “Contra” rebels in Nicaragua.

There were three alternatives:

Simple model: CR - Conservative Rep., LD- Liberal DemocratsSlide31

The first vote was between A and H and the winner to be paired against N.The result was

Consider sophisticated voting (in the last round, insincere voting cannot help, so it must be in the first round)

If H wins the first round, the final outcome is N

But if A wins the first round, the final outcome is A

So the Republicans should vote sincerely for A

LD should vote sincerely for H

But MD should have voted sophisticatedly for ASlide32

Alternatively, we could consider altering the agenda.An appropriate sequential agenda could have produced any one of the alternatives as the winner under sincere voting:Slide33

Impossibility theorem