I have used deterministic optimisation approaches before but never ventured into stochastic optimisation.

In my problem there are a number of decision variables that the optimiser must choose from in order to maximise the objective function. Each decision variable represents a gamble that it will either pay off or return 0.

i.e. choosing between a selection of scratch cards each with a different prize and a different likelihood of winning.

my gut feeling is that I can just put my objective function as

Maximise sum( choose[i] * prize[i] * likelihood[i] for i in scratchcards) within some constraints

where choose is the binary decision variable that is solved for

In my simple (lazy) understanding the above is valid because it represents maximising the expected winnings.

So my question is; do I need to bother with some sort of stochastic optimisation approach? or can I get away with the above?


Whether maximizing expected winnings is an appropriate solution depends on the underlying problem and, crucially, the consumer of the solution (the problem "owner"). Two major considerations jump out at me. One is whether the owner requires a certain minimum payout with a certain probability, in which case you may need to use chance constrained programming (with the minimum payout captured in a chance constraint). For instance, the owner might say "I need to be at least 95% sure of winning at least this amount, or else I don't eat for a week". The other is the utility of winnings, which depends on risk attitude. The value of winnings to the owner may not be a linear function of the dollar amount.

  • $\begingroup$ so to clarify, in this example assume all scratch cards cost the same and the utility of winnings is linear and there are no minimum payouts, just trying to maximise the winnings over a series of repeated games. So assume the number of scratchcards is finite but the number of games is large/infinite. $\endgroup$ Feb 2 '21 at 21:11
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    $\begingroup$ The expected net winnings on a scratchcard is negative; otherwise the lottery would be out of business rather quickly. If someone else is buying the cards for the player, the optimal solution is buy them all. If the player is buying the cards, the optimal solution is to buy none. (This may change if the player's utility function is nonlinear, specifically if the player is risk-seeking as opposed to risk-averse.) $\endgroup$
    – prubin
    Feb 3 '21 at 22:07
  • $\begingroup$ Ok the question is not really about the scratch cards, my application is actually related to trading. The question I'm trying to get across is simply about whether or not it is valid way to handle uncertainty by shoving a likelihood term into the objective and calling it "expected winnings/profit". or do I need to use a scenario tree type approach ? $\endgroup$ Feb 4 '21 at 13:46
  • $\begingroup$ That's really up to the person using the model (the trader). The most definitive thing I can say is that optimizing expected return is a fairly common approach in general. I don't know specifically about trading. $\endgroup$
    – prubin
    Feb 5 '21 at 16:55

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