Goal Gradient.

Our model also implies a goal gradient, or that a reference-dependent player will be more willing to play as her rating approaches the reference point. To see this, compare two players with ratings short of the reference point. For the first player, r₁ + Δ > θ—i.e., a win would push her rating past the reference point. For the second player, r₂ + Δ ≤ θ—i.e., a win would not push her rating past the reference point. The model predicts a goal gradient if player 1’s threshold for playing is lower than player 2’s.

Player 1 plays if α>k−Δ+c(e1∗)−(k+ϵ)⋅F(e1∗), as in Eq. 7, with e1∗ solving the first-order condition in Eq. 4. Player 2 plays if α>k−Δ+c(e2∗)−k⋅F(e2∗), with e2∗ solving the first-order condition in Eq. 2. Hence, player 1’s threshold for playing is lower if (k+ϵ)⋅F(e1∗)−c(e1∗)>k⋅F(e2∗)−c(e2∗), i.e., if the expected net gain from winning (in utiles) is greater for player 1 than for player 2. This condition always holds. The player exerts effort until marginal gains equal marginal costs. Since e1∗>e2∗, player 1 exerts more effort than player 2, implying that net gains are larger for player 1.