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# Wstęp do Teorii Gier PowerPoint PPT Presentation

Wstęp do Teorii Gier. 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)

Wstęp do Teorii Gier

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## Wstęp do TeoriiGier

### 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.

### Eliciting preferences

• Management 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

### The game

• We assume that these utilities are additive (strong assumption)

• We get the following table

• kjh

### Issues

• What 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, enlarge the set of possibilities – brainstorming with LU and management

• Is 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)

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

### Management false utilities

• Suppose, the management misrepresents by doubling negative utilities:

• 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.

### Other cases

• Assume 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)

(3,2) PRCA

In real utilities (3.5,0.5)

PC

(1,0.5)

RCA

In real utilities: (1,2)

(1,1) RC

In real utilities the same (0,0)

(0,0) SQ

### An introduction to N-person games

• Let’s consider a three person 2x2x2 zero-sum game

### Players may want to form coalitions

• Suppose 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.

### Now two remaining possible coalitions

• Rose and Larry against Colin

• Rose and Colin against Larry

### 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:

### 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 other

### Transferable Utility (TU) models

• Von 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.

### Cooperative game with TU

• We 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 here

• Remember going from extensive form game to normal form game, we needed to abstract away specific sequence of moves

• Now in going from a game in normal form to a game in characteristic function form, we abstract away specific strategies

### 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:

• kjhn

### Examples

• N={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 otherwise

• The game is constant-sum and superadditive.

### Elections 1980

• Three candidates:

• Democrat Jimmy Carter,

• Republican Ronald Reagan,

• Independent John Anderson.

### Politics

• In 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 choice

• 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 candidate

• But, 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.

• 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 winning

### 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 Democrats

• 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 A

• 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: