First, consider the problem of optimizingygivenx. This amounts to alinear program:minimizey(Px)subject toeTy= 1y≥0.It is easy to see that the basic feasible solutions of this linear program aregiven bye1, . . . , eM, where eacheiis the vector with all components equal to0 except for theith, which is equal to 1. It follows thatmin{y2<M|ye=1,y≥0}yPx=mini2{1,...,M}(Px)i.This minimal value can also be expressed as the solution to a linear program:maximizevsubject tovePx,wherev2 <is the only decision variable andxis fixed. In particular, theoptimal valuev⇤resulting from this linear program satisfiesv⇤=min{y2<M|ye=1,y≥0}yPx.To determine an optimal strategy for player 1, we find the value ofxthatmaximizesv⇤. In particular, an optimal strategy is delivered by the followinglinear program:maximizevsubject tovePxex= 1x≥0,wherev2 <andx2 <Nare decision variables. An optimal solution to thislinear program provides a stochastic strategyxthat maximizes the payo↵v,

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102assuming that player 2 knows the randomized strategy of player 1 and selectsa payo↵-minimizing counter-strategy. We illustrate application of this linearprogram through a continuation of Example 4.4.2.Example 4.4.2. (linear programming for drug running)To determinean optimal drug running strategy, we formulate the problem in the terms wehave introduced. The drug lord’s strategy is represented as a vectorx2 <3of three probabilities. The first, second, and third components represent theprobabilities that a ship is sent to San Diego, Los Angeles, or San Francisco,respectively. The payo↵is1if a ship gets through, and0otherwise. Hence,the expected payo↵Pijis the probability that a ship gets through if player 1selects decisionjand player 2 selects decisioni. The payo↵matrix is thenP=2642/31111/21111/4375.The optimal strategy for the drug lord is given by a linear program:maximizevsubject tovePxex= 1x≥0.Suppose that the drug lord computes an optimal randomized strategyx⇤by solving the linear program. Over time, as this strategy is used to guideshipments, the drug lord can estimate the Coastguard’s strategyy. Giveny,he may consider adjusting his own strategy in response toy, if that will in-crease expected payo↵. But should it be possible for the drug lord to improvehis expected payo↵after learning the Coastguard’s strategy? Remarkably, ifthe coastguard selects a randomized strategy through an approach analogousto that we have described for the drug lord, neither the drug lord nor theCoastguard should ever need to adjust their strategies.We formalize thisidea in the context of general two-player zero-sum games.Recall from our earlier discussion that player 1 selects a randomized strat-egyx⇤that attains the maximum inmax{x2<N|ex=1,x≥0}min{y2<M|ye=1,y≥0}yPx,and that this can be done by solving a linear programmaximizevsubject tovePxex= 1x≥0.

cBenjamin Van Roy and Kahn Mason103Consider determining a randomized strategy for player 2 through an analo-