Fig. 1. Design and features of the tile game.

a Schematic of the tile game for experiments 1–2. First, a gray tile appears on the screen, and the player must press the spacebar before it disappears. If successful, the tile changes color before transitioning to the jackpot or no jackpot state. We have illustrated the tile game with example values for $p_M\left(hit\right)$ and $p_{E\mid M}\left(jackpot\mid hit\right)$; the full set of values are given in panel b. In this example, there is a 50% chance of activating the tile, $p_M\left(hit\right)=.5$. If the tile is activated, there is a 90% chance of transitioning to the jackpot state, $p_{E\mid M}\left(jackpot\mid hit\right)=.9$, and a 10% chance of transitioning to the no jackpot state, $p_{E\mid M}\left(nojackpot\mid hit\right)=.1$. If the tile is not activated, there is a 10% chance of transitioning to the jackpot state, $p_{E\mid M}\left(jackpot\mid miss\right)=.1$, and a 90% chance of transitioning to the no jackpot state, $p_{E\mid M}\left(nojackpot\mid miss\right)=.9$. b Breakdown of $I\left(M;E\right)$, left, and average number of cents earned per trial (expected value), right, for different values of $p_M\left(hit\right)$ and $p_{E\mid M}\left(jackpot\mid hit\right)$.