Wstęp do Teorii Gier

1. Wstęp do TeoriiGier

2. 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.

4. 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

5. The game • We assume that these utilities are additive (strong assumption) • We get the following table

6. Finding Nash solution • kjh

7. 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

8. Management false utilities • Suppose, the management misrepresents by doubling negative utilities:

9. 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.

10. 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)

11. (3,2) PRCA

12. In real utilities (3.5,0.5) PC (1,0.5) RCA

13. In real utilities: (1,2) (1,1) RC

14. In real utilities the same (0,0) (0,0) SQ

15. An introduction to N-person games • Let’s consider a three person 2x2x2 zero-sum game

17. 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.

18. Now two remaining possible coalitions • Rose and Larry against Colin • Rose and Colin against Larry

19. 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:

20. 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

21. 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.

22. 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

23. 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:

24. Examples • kjhn

25. 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.

26. Elections 1980 • Three candidates: • Democrat Jimmy Carter, • Republican Ronald Reagan, • Independent John Anderson.

27. 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

28. 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.

29. 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

30. 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

31. 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

32. 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:

33. Impossibility theorem