Make Econ Scientific

A Better Motivation Function for ERC

The “motivation function” is an improved utility function in that it takes account of both wealth and relative wealth. But as specified by Bolton and Ockenfels it is not consistent between games (economies) with different numbers of players.

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 = 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:

  1. Consider m() for a specific player i
  2. m(1, 1/2) has a zero derivative w.r.t. sig in a 2-player game with c = 2.
  3. so m(1, 1/3) does not have zero derivative.
  4. But consider the same player in a 3-person game, with c =3, and we find:
  5. m(1, 1/3) has a zero derivative w.r.t. sig.

There seem to be two ways out.

  1. 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).
  2. Define m = m(y, R), where R = n y / c = wealth / (average wealth).

ERC: A Theory of Equity, Reciprocity, and Competition, 2000

3 Responses

  1. re “family of motivation functions” …

    you may find some of the work of B. Mandelbrot
    on multi-fractal systems ( … and the connections
    to general dynamical systems and chaotic attractors)
    both interesting and useful.

    best,

    cdm

    1. I wrote the fastest program for calculating the Mandelbrot set, and handed it to him on a floppy many years ago when he gave a talk at UC Berkeley. Never heard back. The fractals are very cool, but chaos theory seems to have produced mostly chaos, as far as I can tell.

      I agree that human behavior is chaotic, scientific testing will require a far more modest approach for a long time to come. Mandelbrot was anything but modest.

      Thanks much for your interest. –Steve