The evaluation criterion for choices about bets on urn R is straightforward. Starting with the uniform prior
\(\mu _0 = (1/2,1/2)\) over the parameter space
\(\Theta =\left\{ 1/3 , 2/3 \right\} \), the decision-maker uses Bayesian updating to process the information provided by the observed signal history
\(s^t\) to a posterior
\(\mu _t\). This posterior, in turn, is directly associated with a ‘one-step ahead belief’
\(p_t (\cdot ) = \int _\Theta l(\cdot | \theta ) \mathrm{d} \mu _t (\theta )\) on
S about the colour of the next ball. Here,
\(l(\cdot | \theta )\) is the likelihood of a draw
\(s\in S\) given a composition of the urn
\(\theta \), i.e.,
\(l(s=B | \theta ) = \theta \) and
\(l(s=W | \theta ) = 1- \theta \). To rank different bets, the decision-maker compares expected payoffs. For instance, the expected payoff
\(\pi \) of bet
\({x}_{B}^{R} y \) given the one-step ahead belief
\(p_t(\cdot )\) is
$$\begin{aligned} \mathbb {E}^{p_t} \pi (x_B y) = x \, p_t (s=B) + y\, p_t (s=W). \end{aligned}$$
(1)
There are different decision criteria for bets on urn A. The SEU decision-maker forms a unique prior about the composition of the urn and then follows the standard Bayesian updating procedure, so that the decision rule coincides with that in (
1). In the absence of any observations, the uniform distribution is a plausible prior and the betting behaviour regarding urn R and urn A indistinguishable. In contrast to SEU, the intertemporal MEU decision-maker initially holds a
set of priors
\(\mathcal {M}_0\) that is, upon arrival of new information, updated to a set of posteriors
\(\mathcal {M}_t\). This updating process is characterized by MLU; the next subsection explains this process in detail. The set of posteriors
\(\mathcal {M}_t\) is, in turn, associated with a set of ‘one-step ahead beliefs’
\(\mathcal {P}_t(s^t) = \left\{ p_t (\cdot ) = \int _\Theta l(\cdot | \theta ) \mathrm{d} \mu _t (\theta ) \ : \mu _t \in \mathcal {M}_t(s^t)\right\} \) on
S about the colour of the next ball. The MEU decision-maker ranks bets according to maxmin expected utility. Given
\(\mathcal {P}_t(s^t)\), the maxmin expected payoff of bet
\({x}_{B}^{A} y\) is
$$\begin{aligned} \min _{p \in \mathcal {P}_t(s^t)} \mathbb {E}^{p_t} \pi (x_B y ). \end{aligned}$$
(2)
Note that the model presented here is much simpler than the framework of Epstein and Schneider (
2007). Here, the decision task is structurally the same at all points in time. In particular, past choices in no way restrict the remaining decision-tree. In addition, the current choices do not impact the ‘informativeness’ of the next ball drawn; there is no room for experimentation and similar considerations of giving up short-term payoffs for long-term information benefits (see, for instance, Moscarini and Smith
2001). Every decision just focuses on the colour of the next ball and has to be made solely based on what the decision-maker has learned about the composition of the urn. In particular, discounting does not play a role and can be ignored.