Source
Börgers and Sarin, LIT-1505, Section 6 (pp. 19–25 of the 1995 working paper), read in full in NOTE-1149.
What was actually shown
- The rule. A negative payoff takes probability |Ũ|p_j from the strategy played and gives it to the others, in proportion to their probabilities or to one chosen at random in that proportion. The two have the same expected motion.
- The limit. In terms of “benefits”, events that raise j’s probability, dp_j/dt = p_j(Ū_j* − p|Ũ|q) + Ū_j*(σ_j* − p_j), where σ_j* is the probability that j is reinforced and Ū_j* the expected size of its benefit. The first term is a replicator term in benefits, the second the “probability matching force”. The proof is omitted; the authors say they “believe” Norman’s conclusion extends because the irregularities lie only on the boundary (p. 20).
- Cases. All payoffs positive: σ_j* = p_j and the matching term vanishes, giving the replicator equation. All payoffs of one absolute size: the replicator term vanishes. In a one-player problem the choice probability goes to the probability μ of the state in which it pays (Example 2); in a ±0.5 matching pennies the mixed equilibrium is globally stable, which the authors call accidental (p. 25).
- Aspirations. With an aspiration level that tracks experienced payoff, some experiences fall below it, so the matching term is “in most cases unavoidable” (p. 26). That is reported from a companion paper.
What this does not say
- That the process maximizes anything: its limit points are generally not Nash equilibria (p. 25).
- That human choice probability-matches for this reason. The authors cite the psychology only as a known association.
Connections
The record’s common-currency line asks when a negotiation’s choices follow an ordering of its options. This is a mechanism by which a reinforcement learner settles at mixed choice where one option is strictly better on expectation: a departure from maximizing produced by the update rule, not by any failure of the options to be ordered.