Wstęp do Teorii Gier

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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)

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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
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)
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:
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.
• 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: