It is helpful to think about the game as played on a k × k array of lights, each can be either on or off, with switches for each row and each column, such that if a switch of a row or column is pulled then all the lights in that row or column are switched (from on to off or from off to on). Every female player controls a row switch and every male player controls a column switch. The purpose of a female players is to have as many lights ‘on’ in her row, and the purpose of a male player is to have as many lights ‘off’ in his column. In a similar framework, Gale and Berlekamp introduced a two-player zero-sum game with perfect information, where Player 1 chooses the initial configuration of the lights, and then Player 2 chooses which switches to pull. The goal of player 2 is to minimize the number of lights. The following lemma says that there exists some initial configuration of the lights such that whatever switches are performed on the rows, many columns will be unbalanced (i.e. will have much more lights on than off or vice versa). 4.1. Lemma. For sufficiently large k there exists k × k matrix M with entries in { +1 , − 1 } such that (3) # { 1 ≤ j ≤ k : | (¯ xM )$_{j}$ | > √$_{k}$ 20 } > k/ 3 for every row vector ¯ x of length k with entries in { +1 , − 1 } . Proof of Theorem 2.8. By Lemma 4.1 for every sufficiently large k and δ = 20 $_{/}$√$_{k <}$ 30 $_{/}$√$_{n}$ there exits a k × k matrix M with the property (4) # { 1 ≤ j ≤ k : | (¯ xM )$_{j}$ | > 1 δ } > k/ 3 for every strategy profile ¯ x of the females. We claim that if M satisfies (4) then the game admits no 1 / 3-equilibrium. Indeed, fix a strategy profile ¯ x for the females and let ¯ y be a profile such that all the males play 1 / 3–best response to ¯ x . Since by changing his strategy a player inverts the sign of his untrancated payoff, it follows that v$_{j}$ ≥ − 1 / 6 for every male j . In particular, v$_{j}$ ≥ 1 whenever | v$_{j}$ | ≥ 1. 9