it. But his aspirations are themselves evolving with experience, which makes it harder for us to
figure out what will happen.
Suppose, however, that unbeknownst to the agent, each action can yield only two payoffs: a
high one, worth h, and a low one, l, where l < h. (Different actions generate h with different
probabilities, so a genuine choice-problem exists.) Then any aspirational dynamic that is governed
by (A3) will ensure that eventually, no matter what the agent initially aspired to, s/he will come to
regard the h payoff as a ‘success’ and the l payoff as a ‘failure’. That is, the sequence of aspirations
a
1
, a
2
, a
3
, . . . will with probability one get sucked into the (l, h) interval and then stay there. The
following result, which we have proven elsewhere (Bendor, Kumar and Siegel 2004), says this more
precisely.
Proposition 0:
Consider a decision-theoretic problem in which the payoffs are either l or h,
and every feasible action produces either payoff with positive probability. If aspirations adjust via
(A3) then the following conclusions hold.
• (i) If a
t
∈ (l, h) then with probability one a
t
∈ (l, h) for all t > t .
• (ii) Suppose aspirations start outside (l, h): either a
0
≤ l or a
0
≥ h. Then a
t
moves mono-
tonically toward (l, h) and is absorbed into that interval with probability one as t → ∞.
Under these assumptions, decision makers whose aspirations adjust in accord with (A3) will
eventually become dissatisfied with l and so will become less inclined to use an action that just
delivered that payoff, and they will become more disposed to an action which has just produced
the h-payoff. Thus, proposition 0 provides an analytical warrant for suppressing aspirations from
models in which payoffs are binary.
Let us transfer this lesson to our analytical model of voting. Here we assume that governments
generate binary payoffs for citizens. (It is natural to posit that a Republican administration is more
likely to produce h’s for conservative voters than for liberal ones, and vice versa for a Democratic
administration. We will discuss these properties in detail shortly.) Proposition 0 implies that if
citizens adjust aspirations in ways consistent with (A3) then eventually their aspirations will be in
(l, h); hence, they will come to regard h’s as satisfying and l’s as dissatisfying. So if they adjust
9
Convention