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Wstęp do Teorii Gier

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Wstęp do TeoriiGier

- 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.

- 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

- Indifferent between $0.67 raise and granting pension benefits
- Management cardinal utility
- Labor union cardinal preferences

- We assume that these utilities are additive (strong assumption)
- We get the following table

- 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

- 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.

- 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

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

- 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.

- Rose and Larry against Colin
- Rose and Colin against Larry

- 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:

- 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

- 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.

- 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

- 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.

- 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

- 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

- 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: