Wst p do teorii gier
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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)

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Wstęp do Teorii Gier

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Wst p do teorii gier

Wstęp do TeoriiGier

Labour union vs factory management

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

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

The game

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

  • We get the following table

Finding nash solution

Finding Nash solution

  • kjh



  • 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)

  • 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 alternatives

Management false utilities

Management false utilities

  • Suppose, the management misrepresents by doubling negative utilities:

Wst p do teorii gier

  • 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

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)

Wst p do teorii gier

(3,2) PRCA

Wst p do teorii gier

In real utilities (3.5,0.5)




Wst p do teorii gier

In real utilities: (1,2)

(1,1) RC

Wst p do teorii gier

In real utilities the same (0,0)

(0,0) SQ

An introduction to n person games

An introduction to N-person games

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

Players may want to form coalitions

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

Now two remaining possible coalitions

  • Rose and Larry against Colin

  • Rose and Colin against Larry

Which coalition will form

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 form1

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

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

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

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



  • 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

Elections 1980

  • Three candidates:

    • Democrat Jimmy Carter,

    • Republican Ronald Reagan,

    • Independent John Anderson.



  • 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

Wst p do teorii gier

  • 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.

Wst p do teorii gier

  • 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

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

Wst p do teorii gier

  • 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

Wst p do teorii gier

  • 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:

Impossibility theorem

Impossibility theorem

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