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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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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
issues
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)
  • 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:
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.
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)
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:
examples1
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
Elections 1980
  • Three candidates:
    • Democrat Jimmy Carter,
    • Republican Ronald Reagan,
    • Independent John Anderson.
politics
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
slide28

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.
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 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
slide31

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
slide32

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:
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