White Oak Intelligence Posted on May 31 • Originally published at whiteoakintel.com The Do-Over Game: Nash Equilibrium at the Golden Ratio # algorithms # computerscience # science In This Article The Question Why 0.50 Is Not the Answer Modeling Your Final Draw Distribution The Indifference Condition for Nash Equilibrium Solving for t*: The Golden Ratio Appears Verifying the Nash Equilibrium Python Simulation and Win Probability Curve Business Application: Optimal Stopping in M&A and Hiring The Question Two players each draw a single number uniformly at random from the interval . After seeing their own draw, each player independently decides whether to redraw — replacing their current number with a fresh uniform draw from — or to keep what they have. A player who redraws must keep the second draw regardless of its value. After both players have finalized their numbers, the player with the higher number wins. Ties are broken arbitrarily (say, in favor of Player 2). Both players make their redraw decision simultaneously and independently. Each is trying to maximize their probability of winning. What is the optimal threshold strategy, and what is the equilibrium threshold value? A threshold strategy is a strategy of the form: "Redraw if my first draw is below ; keep if it is at or above ." We will show that the unique symmetric Nash equilibrium is a threshold strategy, and the threshold is — the reciprocal of the golden ratio. This result appears in quantitative interviews at Jane Street, Citadel, and Goldman Sachs, and it is one of the most striking instances of a famous irrational constant appearing as the solution to a game-theoretic fixed-point problem. Why 0.50 Is Not the Answer The naive threshold is : "If I drew below the median, I am below average, so I should redraw." This reasoning has the right structure — using a threshold strategy — but the wrong threshold. The flaw is that it treats the optimal threshold as a purely individual decision problem, ig
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