Bolton and Ockenfels ERC paper replaces the standard utility function with a motivation function: m = m(y, sig), where y is player i’s wealth and sig is y / (total wealth = c).
When a player has average wealth, sig = 1/n, where n = # of players.
Assumption 3 states that, for any y, dm/dsig = 0 for sig = 1/n. And the second partial of m w.r.t. sig < 0 for all sig. Hence m(y,sig) is maximized for any y, by sig=1/n.
This appears to be contradictory:
- Consider m() for a specific player i
- m(1, 1/2) has a zero derivative w.r.t. sig in a 2-player game with c = 2.
- so m(1, 1/3) does not have zero derivative.
- But consider the same player in a 3-person game, with c =3, and we find:
- m(1, 1/3) has a zero derivative w.r.t. sig.
There seem to be two ways out.
- Assume that a player is described by a family of motivation functions, one for each number of players. I.e., define: m = m(y, sig, n).
- Define m = m(y, R), where R = n y / c = wealth / (average wealth).