A proportional rewards game: Competition among heterogeneous actors*

After a first draft of the paper was finished, the second author died after a long and severe illness. The first author acknowledges the support from the Royal Netherlands Academy of Arts and Sciences (KNAW). Requests for reprints and comments should be directed to the address above.

We analyze a non‐cooperative game for pure distribution settings. A group of n actors competes for a single prize. The probability to win the price can be influenced by making an investment. We assume that the winning probabilities of actors are proportional to their investments. The costs of one unit of investment may vary between actors. A low cost of investment can be interpreted as high productivity in this competition. We call an actor active if he makes a positive investment. The game has a unique, inefficient Nash‐equilibrium. In equilibrium the active actors are remarkably homogeneous with respect to their costs/benefits. The stability of the number of active actors under small changes in the costs of investing, and under the introduction or removal of actors is examined. Under mild conditions, if the number n of actors increases, the number of active actors becomes 2 or the proportion of active actors tends to zero at the rate n ^‐1/2.

Notes

After a first draft of the paper was finished, the second author died after a long and severe illness. The first author acknowledges the support from the Royal Netherlands Academy of Arts and Sciences (KNAW). Requests for reprints and comments should be directed to the address above.