Bargaining solutions without the expected utility hypothesis.

*(English)*Zbl 0776.90094It is well-known in game theory that reasonable axioms that describe human preferences lead to the notion of utility (or expected utility) \(u(A)\) such that \(A\) is preferable to \(B\) iff \(u(A)>u(B)\) and the utility of a lottery \(L\) in which a person gets \(A_ 1\) with probability \(p_ 1,\dots,A_ n\) with probability \(p_ n\) is equal to the mathematical expectation of utilities \(u(L)=\sum p_ i u(A_ i)\) (this equality is why \(u\) is called expected utility). Preferences determine this function \(u\) uniquely modulo some linear transformation \(u\to Au+B\).

Nash proved in 1951 that if we formulate reasonable axioms on what a bargaining solution may look like, then for 2-person bargaining we get a unique results that satisfies these axioms: namely, the alternative \(A\) for which \((u_ 1(A)-u_ 1(S))(u_ 2(A)-u_ 2(S))\to\max\), where \(u_ i\) are utilities of the two players, and \(S\) is a status quo situation. At first glance, it may seem like business bargaining is no more necessary in the real world. Alas, it is still necessary. First, people’s preferences are not always rational (and hence, are not exactly described by utilities). Second, bargaining is not always rational, and hence Nash’s solution is not always applicable. The authors try to generalize Nash’s solution to get both “irrationalities” into consideration.

First, it has been shown that in some cases, if we still impose some rationality restrictions on a person’s preferences, we can still get a function \(u\) that is determined uniquely modulo linear transformation (this function \(u\) will not necessarily be the expected value). In this case, axioms similar to Nash’s lead to a similar solution \((u_ 1(A)- u_ 1(S))(u_ 2(A)-u_ 2(S))\to\max\) (with non-expected utilities \(u_ i)\). A more general case is when such a utility function \(u\) exists only locally, when we compare sufficiently close possibilities. In this case, the authors propose to do the following: Take a neighborhood of a status quo point \(S\) in which utilities are defined, and find the best solution \(S_ 1\) (in Nash’s sense) in this neighborhood. Then, start with \(S_ 1\), find a neighborhood of \(S_ 1\) in which utilities are defined, and find the best solution \(S_ 2\) in that neighborhood, etc. We continue this step-by-step process until we encounter a solution \(S_ m\) that cannot be thus improved.

Similar ideas are proposed for the case when we use a weaker version of Nash’s axioms (that in case of expected utility leads to Kalai- Smorodinsky solution).

Nash proved in 1951 that if we formulate reasonable axioms on what a bargaining solution may look like, then for 2-person bargaining we get a unique results that satisfies these axioms: namely, the alternative \(A\) for which \((u_ 1(A)-u_ 1(S))(u_ 2(A)-u_ 2(S))\to\max\), where \(u_ i\) are utilities of the two players, and \(S\) is a status quo situation. At first glance, it may seem like business bargaining is no more necessary in the real world. Alas, it is still necessary. First, people’s preferences are not always rational (and hence, are not exactly described by utilities). Second, bargaining is not always rational, and hence Nash’s solution is not always applicable. The authors try to generalize Nash’s solution to get both “irrationalities” into consideration.

First, it has been shown that in some cases, if we still impose some rationality restrictions on a person’s preferences, we can still get a function \(u\) that is determined uniquely modulo linear transformation (this function \(u\) will not necessarily be the expected value). In this case, axioms similar to Nash’s lead to a similar solution \((u_ 1(A)- u_ 1(S))(u_ 2(A)-u_ 2(S))\to\max\) (with non-expected utilities \(u_ i)\). A more general case is when such a utility function \(u\) exists only locally, when we compare sufficiently close possibilities. In this case, the authors propose to do the following: Take a neighborhood of a status quo point \(S\) in which utilities are defined, and find the best solution \(S_ 1\) (in Nash’s sense) in this neighborhood. Then, start with \(S_ 1\), find a neighborhood of \(S_ 1\) in which utilities are defined, and find the best solution \(S_ 2\) in that neighborhood, etc. We continue this step-by-step process until we encounter a solution \(S_ m\) that cannot be thus improved.

Similar ideas are proposed for the case when we use a weaker version of Nash’s axioms (that in case of expected utility leads to Kalai- Smorodinsky solution).

Reviewer: O.M.Kosheleva (El Paso)

##### MSC:

91A12 | Cooperative games |