Nash equilibrium allocation (game theory)

用通俗的话解释

Imagine an opponent who, after seeing your portfolio, chooses which of the past weeks will happen again. In this zero-sum game the best you can do is a mixed strategy whose worst week is as good as possible: the minimax, or Nash equilibrium, portfolio. It is the most pessimistic model in the optimiser.

计算方法

The scenarios are the past weekly returns (5 trading days). The investor chooses a mix of tickers, nature a mix of weeks. Both adapt by multiplicative weights over 4,000 rounds: the investor raises the weight of tickers that did well against nature's choice, nature raises the weight of weeks that hurt the investor. The time averages converge to the equilibrium. Long only and fully invested.

value = max_w min_s Σᵢ wᵢ R_{s,i}
pₜ₊₁ ∝ pₜ × exp(η Rᵀqₜ)     qₜ₊₁ ∝ qₜ × exp(−η R pₜ)

如何使用

• As a stress-minded benchmark: what if the worst of history repeats?
• Compare its game value (the guaranteed weekly result) with other models' worst week.

得出结论

• A game value below zero means every mix had some losing week; it tells you the least bad one.
• It favours whatever was most stable week to week.

注意事项

• Very conservative and blind to returns on all the other weeks.
• With more scenarios (longer history) the worst cases get worse.

一点历史

John von Neumann proved the minimax theorem for zero-sum games in 1928; John Nash generalised equilibrium in 1950. Freund and Schapire (1999) showed that multiplicative-weight learning, used here, finds such equilibria.