Suppose a betting game has three possible outcomes. If a gambler bets on outcome i, then he receives a net gain of d_i dollars for one dollar betted, i = 1, 2, 3. The payoff matrix thus takes the form (consideration of discounting is not necessary in a betting game)

Find the condition on d_i such that a risk neutral probability measure exists for the above betting game (visualized as an investment model).

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S(1; 2): = d₁ + 1 0 0 0 d₂ + 1 0 0 0 d3 + 1