Fig. 6. Design of experiment 3.

In experiment 3, each version of the tile game shared the following features: (i) the tile glowed when activated (instead of changing colors, as in experiments 1 and 2), (ii) $p_M\left(hit\right)$ was always set to .5, and (iii) $p_{E\mid M}\left(jackpot\mid hit\right)$ was always set to 1. a Mixture game. Missing the tile results in a 50% chance of the tile turning red, indicating a $0.02 loss (i.e., a deduction of $0.02 from the player’s bonus fund), and a 50% chance of the tile turning gray (indicating no jackpot); M has three possible values (hit, gray, and red) and E has three possible values (jackpot, no jackpot, and penalty). b Punish game. Missing the tile results in a 100% chance of the tile turning red (indicating a $0.02 less); M has two possible values (hit and red) and E has two possible values (jackpot and penalty). c Neutral game. Missing the tile results in a 100% chance of the tile turning gray, indicating no jackpot; M has two possible values (hit and gray) and E has two possible values (jackpot and no jackpot). d Key statistics associated with each version of the game. The values of $I\left(M;E\right)$, expected value (EV), marginal value ($\Delta V$), and the correlation between $M$ and $E$ (${\mathrm{\phi }}_{\mathrm{c}}\left(M,E\right)$) were computed analytically. The values for temporal difference prediction error ($\mathrm{\delta }\left(M\right)$) and value of information (VOI) correspond to the average output of 1,000 simulations. The values of skill-challenge balance (SCB) and controllability (CTRL) are the empirical means and standard errors obtained from self-report measures of the corresponding constructs. The largest value of each variable is highlighted in gray.