Zero Sum Games and Nash Equilibrium for Traders

Nash equilibrium describes the strategy a trader cannot improve on alone, and zero sum toy games teach that logic directly.

A zero sum game is one where one participant's gain is exactly another participant's loss, and a Nash equilibrium is the set of strategies where no participant can improve their own outcome by changing their strategy alone, given what everyone else is doing. Traders encounter both ideas directly whenever they compete for the same fill against other market participants who are reasoning about the same information.

Trading itself is not inherently zero sum, since two counterparties can both be better off relative to their own objectives, one gaining liquidity and the other gaining a favorable price. But specific competitive situations inside trading are zero sum: two market makers quoting for the same order flow are competing for a fixed prize, since only one of them wins the trade, and whichever one wins takes the fill the other one wanted.

Interviewers use small, literal zero sum games such as matching pennies or a simplified rock paper scissors variant to test whether you can find a stable strategy, because the reasoning behind those toy games is identical to the reasoning behind quoting against a competing market maker.

In matching pennies, two players each show a penny as heads or tails. Player one wins if the pennies match, player two wins if they differ. If player one always played heads, player two would learn this and always play tails, winning every round. The only stable strategy is for each player to randomize 50 percent heads and 50 percent tails, which makes the opponent indifferent between their own two choices and removes any pattern for them to exploit.

This is the Nash equilibrium of the game: neither player can improve their expected outcome by unilaterally switching strategies, since any fixed pattern, once discovered, gets exploited down to a losing one. The expected value of the game to each player at equilibrium is exactly zero, which is the defining feature of a fair zero sum game with no informational edge on either side.