Game Theory Lecture 14 Labour union vs factory management •  The management of a factory is nego8a8ng a new contract with the union represen8ng 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 (reduc8on necessary) (A) •  You have been called as an arbitrator. Elici8ng preferences •  Management ordinal preferences •  Further ques8ons: –  Indifferent between $0.67 raise and gran8ng pension benefits •  0.67R=P, hence P=‐2 and R=‐3 –  Willing to trade off a full raise and half of pension benefits for elimina8on of the coffee break •  R+0.5P=‐C, hence C=4 •  Management cardinal u8lity •  Labor union cardinal preferences The game •  We assume that these u8li8es are addi8ve (strong assump8on) •  We get the following table Finding Nash solu8on •  kjh Issues •  What if the Nash point is a mixed outcome? E.g. (2,2½)= ¼PRC+ ¾PRCA. –  How to give ¾ of the automa8on? –  Possibili8es: grant automa8on 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 beeer, enlarge the set of possibili8es – brainstorming with LU and management •  Is the present situa8on a good SQ? –  Real nego8a8on ohen take place in an atmosphere of threats, with talks of strikes and lockout (each side tries to push SQ in her more favorable direc8on) •  What about false informa8on about u8li8es given by each side? –  E.g. correct scaling for posi8ve and nega8ve u8li8es separately, but to misrepresent the “trading off” of the alterna8ves Management false u8li8es •  Suppose, the management misrepresents by doubling nega8ve u8li8es: The new Nash point is at (1,½) It could be implemented as: •  ½PC + ½RCA. In the honest u8li8es this point corresponds to (3½,½) ‐not Pareto op8mal, but beeer for the management than (3,2) •  Or ¾PC + ¼C. In the honest u8li8es 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 nega8ve payoffs –  The solu8on RC (LU does not profit) •  Assume that both lie and double their nega8ve u8li8es –  The solu8on SQ!!! (No profitable trade at all) An introduc8on to N‐person games •  Let’s consider a three person 2x2x2 zero‐sum game Players may want to form coali8ons •  Suppose Colin and Larry form a coali8on 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 coali8ons •  Rose and Larry against Colin •  Rose and Colin against Larry Which coali8on will form? •  How the coali8on 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 coali8on! –  Colin though not very well off, is s8ll beeer off than when facing Rose and Larry against him. •  The rest of the calcula8ons is as follows: Which coali8on will form? •  For each player, find that player’s preferred coali8on partner. •  For instance Rose would prefer Colin as she wins 2.12 with him compared to only 2.00 in coali8on with Larry. •  Similarly Colin’s preferred coali8on partner is Larry •  Larry’s preferred coali8on partner is Colin. •  So Larry and Colin would form a coali8on! •  Unfortunately, it may happen that no pair of players prefer each other Transferable U8lity (TU) models •  Von Neumann and Morgenstern made an addi8onal assump8on: they allowed sidepayments between players •  For example Rose could offer Colin a sidepayment of 0.1 to join in a coali8on with her – effec8ve payoffs (2.02,‐0.59,‐1.43) –  This coali8on is more aerac8ve to Colin than Colin‐Larry coali8on •  The Assump8on that sidepayments are possible is very strong: –  It means, that u8lity is transferable between players. –  It also means, that u8lity is comparable btw. players. –  Reasonable when there is a medium of exchange such as money. Coopera8ve game with TU •  We assume that: –  Players can communicate and form coali8ons with other players, and –  Players can make sidepayments to other players •  Major ques0ons: –  Which coali8ons should form? –  How should a coali8on which forms divide its winnings among its members? •  Specific strategy of how to achieve these goals is not of par8cular 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 characteris0c func0on form, we abstract away specific strategies Characteris8c func8on •  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 rela8on: Examples •  kjhn 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 cona8ns at least 2/3 of both the House and the Senate. •  v(S)=0 otherwise •  The game is constant‐sum and superaddi8ve. Elec8ons 1980 •  Three candidates: –  Democrat Jimmy Carter, –  Republican Ronald Reagan, –  Independent John Anderson. Poli8cs •  In the summer before the elec8on, 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 conserva8ve than Anderson and Anderson was more conserva8ve than Carter. –  Assump8on: 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 op8mal 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 leh •  Carter and Anderson voters could improve by vo8ng 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 Representa8ves defeated a plan to provide humanitarian aid to the US backed “Contra” rebels in Nicaragua. •  There were three alterna8ves: •  Simple model: CR ‐ Conserva8ve Rep., LD‐ Liberal Democrats •  The first vote was between A and H and the winner to be paired against N. •  The result was •  Consider sophis8cated vo8ng (in the last round, insincere vo8ng cannot help, so it must be in the first round) – 
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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 sophis8catedly for A •  Alterna8vely, we could consider altering the agenda. –  An appropriate sequen8al agenda could have produced any one of the alterna8ves as the winner under sincere vo8ng: Impossibility theorem