BoundedRationality_TheAdaptiveToolbox.pdf

314 WulfAlters
Nash problem with conflict point: This uses the same problem as described
above, but if there is no agreement, players receive the conflict payoff
(q,..., cn). In the two-person case, it is easy to present the preference criterion in
the form P"(i) < > P"(j) as u(xt) - u(yt) + w(cz) < > u(y) - u(xj) + u(c), which
adds the terms u(cj) and u(c) on both sides of the inequation obtained from the
paragraph above. The form P / (x)Ffy) is obtained as u(xt) + u(x) +
u(ct) < > u(yt) + u(yA + u(cX where every alternative is supported by the conflict
payoff of the player who supports it. This two-person approach cannot be
easily transfered to the ^-person case, since the idea of adding the values w(c^) of
the players / who prefer the respective alternative to the respective side of the inequality,
does not need to fulfill the fifth principle of aggregation (see above).
Specifically, the fifth principle cannot be fulfilled when the number of players is
odd.
Harsanyi—Selten problem: As above, but here the players must decide simultaneously,
each player for one of the alternatives. If both players select the same
alternative, they receive the corresponding payoff. If both of them select their
best alternative, they receive the conflict payoff (q, c2). If both of them select
the best alternative of the respective other player, they receive the
miscoordination payoff (m1? m2). The preference criterion in the form
P"(i) P"{j) is obtained as u(xx) — uiy^) + u{cx) — u{mx) ufy^) — u(x2) +
u(c2) — u(m2).
Kalai-Smorodinsky problem: As the Nash problem with conflict point, but
here there is also a "bliss"or "utopia" point (61? b2), which has the property that
agreement between subjects (Z?j, b2) would be the adequate payoff pair if the
sum bx + b2 could be paid (bt > xt, and bi > yt for both players). In this case the
preference criterion can be presented as u(bi) — u(x^) + u(x2) — u(c2) < >
u(b2) - u(y2) + u(y {) - u(c j). The terms on the two sides of the inequation can be
interpreted as the concessions of the players. At point (x^x2), player 1 has made
the concession u(bx) - u(xx) on her payoff scale by reducing her payoff below
the bliss point; moreover, she made the concession u(x2) — u(c2) by increasing
the opponent's payoff above the lowest possible level c2. A similar description
holds for the concession of player 2 in (y1? y2). The criterion measures which of
the two players made the higher concession.
Optimal solutions: The criteria above can also be used to define optimal solutions
of a set of alternatives (which is usually assumed as convex): an alternative
is optimal with respect to a set of alternatives, if it is preferred to all other alternatives
of the set. For the Nash problem, the optimal solution is characterized by
u{xx) + ... + u(xn) = maxl. For the Kalai—Smorodinsky problem, it is the payoff
pair (JCJ, x2) for which u(xi) — u(x2) is nearest to (u(bx)- u(b2) + (u(c{) —
u(c2))/2.
Recent studies by Vogt applied the principles of utility aggregation to ultimatum
games, certain reciprocity games, and certain sequential combinations of
these two types, including the principal agent conflict. For the analysis of these
games he added an additional principle of aggregation: