Portfolio construction
Nash equilibrium allocation (game theory)
Treats investing as a game against the market, which picks the worst past week for you; the result is the mix with the best worst-case outcome.
The idea in plain words
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.
How it is calculated
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)How to use it
- 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.
Drawing conclusions
- 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.
Example
Running example (used in every portfolio guide): US500, GER40, gold (XAUUSD) and EURUSD, two years of daily data. Expected returns 8%, 7%, 7% and 1% a year; volatilities 18%, 20%, 15% and 7%. US500 and GER40 move together (correlation about 0.7); gold hardly moves with the indices (about 0.04). Nash puts 97% in EURUSD and about 1% in each of the others: the worst week of that mix was −2.4%, the best worst week achievable. Return 1.2%: the cost of total pessimism.
The example's inputs are illustrative; the weights and figures are what our optimiser returns for them.
Watch out for
- Very conservative and blind to returns on all the other weeks.
- With more scenarios (longer history) the worst cases get worse.
A little history
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.
Educational content only. No portfolio model knows the future: every one of them works on estimates from the past, and nothing here is investment advice.