TWO COIN MORRAThis game is played bytwo players,RandC.Eachplayer hides either one or two silverdollars in his/her hand. Simultaneously,eachplayer guesses howmany coins the other playeris holding. IfRguesses correctly andCdoes not, thenCpaysRan amount of money equalto thetotalnumber of dollars concealed bybothplayers. IfCguesses correctly andRdoesnot, thenRpaysCan amountofmoneyequaltothetotalnumber of dollars concealed bybothplayers. If both players guess correctly or incorrectly, no money exchanges hands.Clearly,each player must decide howmany coins to hide and what number to guess. Wewill use the notation (1,2) to mean that a player hides 1 dollar and guesses \2." Wecanrepresent the possible outcomes of a round of playbyapayomatrix, indicating howmuchRwill receive fromCgiven the strategies followed byeach player. A negativenumber meansthatCreceives money fromR.C(1,1) (1,2) (2,1) (2,2)R(1,1)(1,2)(2,1)(2,2)0,230200,3,300403,40This is an example of anite two-person zero-sum game.\Finite" refers to the fact thateach player has a nite number of strategies. \Zero-sum" refers to the fact that what oneplayer gains in wealth, the other loses.SupposeRdecides to use only strategy (1,2). This is an example of apurestrategy.Sincethe minimum entry in that rowis,2,Rcan guarantee losing no more than $2 p er round,and this will happen ifCconsistently uses strategy (1,1). But what ifRdecides to useeither strategy (1,2) or (2,1), each half of the time, but randomly mixed so that there is nodetectable pattern? For example, he could ip a coin to decide which of the two strategiesto use. This is an example of amixedstrategy.To calculate the expected outcome, wemixthe second and third rows bymultiplying them eachby 1/2 and adding them together. Theresult is12h,2 0 0 3i+12h3 0 0,4ikh120 0,12i:1Since the minimum entry is,1=2,Rcan exp ect to lose no more than half a dollar p er roundon the average, and he can exp ect this to happen ifCconsistently chooses strategy (2,2).CanRdo better than this? IfRuses each strategy 1/4 of the time, thenRmixes hisrows bymultiplying each of them by 1/4 and adding them together, giving14h0 2,3 0i+14h,2 0 0 3i+14h3 0 0,4i+14h0,3 4 0ikh14,1414,14i:Since the minimum entry is,1=4,Rwould expect to lose no more than a quarter dollar perround on the average, and he can exp ect this to happ en ifCconsistently uses only strategies(1,2) and (2,2).Problem:Try to nd a mixed strategy forRwhichiseven b etter. In particular, try tond nonnegativenumbersp1,p2,p3,p4that sum to one such that the minimum entry inp1h0 2,3 0i+p2h,2 0 0 3i+p3h3 0 0,4i+p4h0,3 4 0iis at least zero. Can you nd a mixture where the minimum entry is larger than zero?C's strategies can be studied in a similar manner. For example, supp oseCdecides to useonly strategy (2,2). The numbers in the matrix represent amounts thatCpaysR,soCisinterested in looking at the maximum entry to see how badly he will do. Since the maximumentry in column 4 is 3,Ccan expect to lose at most $3 p er round, and this happ ens ifRconsistently uses strategy (1,2). IfCuses each of his strategies a fourth of the time, wemust2mix the columns bymultiplying eachof themby 1/4 and adding them together. This yields14266640,23037775+1426664200,337775+1426664,300437775+142666403,4037775=26664,1=41=4,1=41=437775:Since the maximum entry is 1/4, by using this mixed strategy,Ccan exp ect to lose no morethan a quarter dollar per round on the average, and he can exp ect this ifRconsistently usesonly strategies (1,2) and (2,2).Of course, by symmetry,we can exp ect the best strategy forRto b e the same as thebest strategy forC.Here is how to nd the best strategy forRby linear programming: Find the optimalsolution tomaxzs.t.p1+p2+p3+p4=1z0p1,2p2+3p3+0p4z2p1+0p2+0p3,3p4z,3p1+0p2+0p3+4p4z0p1+3p2,4p3+0p4p1;p2;p3;p40One (the only?) optimal solution isp2=3=5,p3=2=5, with a value of 0. By symmetry,this will also b eC's optimal strategy.Problem:What ifRandCdo not announce their guesses simultaneously,butRguessesrst,Cguesses second, and then b oth op en their hands to see who is correct? Do es the factthatRguesses b eforeCgiveCanyadvantage, even thoughCstill does not knowhow manycoinsRis hiding? Try solving this problem using linear