I contemplate an action on my friend and the resulting consequences \(\{E_i\}\). Let \(X\) be a gamble whose payoff depends on these consequences, and let \(P(X)\) be my prevision for \(X\), that is, my fair price for \(X\). Letting \(x_i\) be the payoff conditional on outcome \(E_i\), by the linearity of previsions, we have \[\tag{0.1}\begin{aligned}P(X) = \sum_i x_i P(E_i) .\end{aligned}\]
I may also contemplate asking my friend: what fair price do you think I will assign to \(X\)? This is an action I can take on my friend, and their reply is a consequence: \(X_0\). The principle of interpersonal indifference says I ought to adopt \[\tag{0.2}\begin{aligned}P(X|P_F(X) = X_0) = X_0,\end{aligned}\]
that is, conditional on the friend’s announcement that they believe I will assign a fair price \(X_0\) to \(X\), I myself ought to value \(X\) at \(X_0\). By the law of total prevision, \[\tag{0.3}\begin{aligned}P(X) &= \sum_{ij} x_i P\big(E_i|P_F(X) = X_j\big)P\big(P_F(X) = X_j\big)\end{aligned}\] \[\tag{0.4}\begin{aligned}&= \sum_j P\big(X|P_F(X) = X_j\big)P\big(P_F(X) = X_j\big)\end{aligned}\] \[\tag{0.5}\begin{aligned}&= \sum_j X_j P\big(P_F(X) = X_j\big).\end{aligned}\]
Therefore before I make any action, if I adopt the principle of interpersonal indifference, I should assign a prevision \(P(X)\) which is a mixture of possible fair prices \(X_j\) weighted by my probability that my friend will announce they believe I will assign prevision \(X_j\) to my action on them.
Treating \(P_F(X)\) itself as a gamble which pays \(X_j\) if my friend announces it, then we have \(P(P_F(X)) = \sum_j X_j P\big(P_F(X) = X_j\big)\) so that \[\tag{0.6}\begin{aligned}P(X) = P(P_F(X)).\end{aligned}\]
Thus I regard two gambles as interchangeable: a) the original gamble \(X\) whose payoff depends on the consequences \(\{E_i\}\); and b) the gamble \(P_F(X)\), whose payoff depends on the price my friend announces as the price they think I would assign to \(X\). In this sense, I believe my friend understands me: I am willing to use their view of my own judgments as a substitute for those judgments themselves.
In the simplest case, we can consider the gamble which pays \(\$ 1\) if \(E_i\). Then interpersonal indifference amounts to \(P(E_i|P_F(E_i) = p_0) = p_0\) where \(p_0\) is a probability, and \[\tag{0.7}\begin{aligned}P(E_i) = \sum_j p_j P(P_F(E_i) = p_j).\end{aligned}\]
Now suppose the friend actually announces \(P_F(E_i)=p_0=1\) upon which I adopt \(P(E_i)=1\). (This is a slightly stronger statement of interpersonal indifference.) But then when I actually act on the friend according to \(\{E_i\}\), they freely answer \(E_j\) for \(j \neq i\). This is a problem! Because I essentially allow my friend to freely set my probabilities for \(\{E_i\}\) as well as to freely pick an answer \(E_i\), I open myself up to a loss. But this is the essence of interpersonal indifference: I open myself up to risk by trusting my friend. In other words, I trust that my friend will treat my losses as if they were their own.
Principles:
Understanding: I accept my friend’s view of my judgment as a substitute for my own judgment.
Trust: I believe my friend will not knowingly announce a value that misrepresents the judgment they sincerely expect me to make.
Solidarity: I believe my friend will not exploit the vulnerability produced by my adoption of their announcement, because they give my losses weight comparable to their own.