Stochastic Programming - Index of

156 STOCHASTIC PROGRAMMING
an exercise.
Example 2.5 Let us conclude this section with another similar example. You
are to throw a die twice, and you will win 1 if you can guess the total number
of eyes from these two throws. The optimal guess is 7 (if you did not know
that already, check it out!), and that gives you a chance of winning of 1 6 .So
the expected win is also 1 6 .
Now, you are offered to pay for knowing the result of the first throw. How
much will you pay (or alternatively, what is the EVPI for the first throw) A
close examination shows that knowing the result of the first throw does not
help at all. Even if you knew, guessing a total of 7 is still optimal (but that
is no longer a unique optimal solution), and the probability that that will
happen is still 1 6
. Hence, the EVPI for the first stage is zero.
Alternatively, you are offered to pay for learning the value of both throws
before “guessing”. In that case you will of course make a correct guess, and
be certain of winning one. Therefore the expected gain has increased from 1 6
to 1, so the EVPI for knowing the value of both random variables is 5 6 . ✷
As you see, EVPI is not one number for a stochastic program, but can
be calculated for any combination of random variables. If only one number is
given, it usually means the value of learning everything, in contrast to knowing
nothing.
References
[1] Bellman R. (1957) Dynamic Programming. Princeton University Press,
Princeton, New Jersey.
[2] Helgason T. and Wallace S. W. (1991) Approximate scenario solutions in
the progressive hedging algorithm. Ann. Oper. Res. 31: 425–444.
[3] Howard R. A. (1960) Dynamic Programming and Markov Processes. MIT
Press, Cambridge, Massachusetts.
[4] Nemhauser G. L. (1966) Dynamic Programming. John Wiley & Sons, New
York.
[5] Rockafellar R. T. and Wets R. J.-B. (1991) Scenarios and policy aggregation
in optimization under uncertainty. Math. Oper. Res. 16: 119–147.
[6] Schaefer M. B. (1954) Some aspects of the dynamics of populations
important to the management of the commercial marine fisheries. Inter-
Am. Trop. Tuna Comm. Bull. 1: 27–56.
[7] Wallace S. W. and Helgason T. (1991) Structural properties of the
progressive hedging algorithm. Ann. Oper. Res. 31: 445–456.
[8] Watson S. R. and Buede D. M. (1987) Decision Synthesis. The Principles