The Prediction value

We introduce the prediction value (PV) as a measure of players' informational importance in probabilistic TU games. The latter combine a standard TU game and a probability distribution over the set of coalitions. Player $i$'s prediction value equals the difference between the conditional expectations of $v(S)$ when $i$ cooperates or not. We characterize the prediction value as a special member of the class of (extended) values which satisfy anonymity, linearity and a consistency property. Every $n$-player binomial semivalue coincides with the PV for a particular family of probability distributions over coalitions. The PV can thus be regarded as a power index in specific cases. Conversely, some semivalues -- including the Banzhaf but not the Shapley value -- can be interpreted in terms of informational importance.