INTERVAL GAMES. and player 2 selects 1, then player 2 would give player 1 a payoff of, 1) = 0.
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1 INTERVAL GAMES ANTHONY MENDES Let I ad I 2 be itervals of real umbers. A iterval game is played i this way: player secretly selects x I ad player 2 secretly ad idepedetly selects y I 2. After x ad y are revealed, payoffs are give by some predetermied fuctio with domai I I 2. The payoff fuctio for a zero sum iterval game is a fuctio A : I I 2 R. This is iterpreted to mea that the secod player gives the first player a payoff of A(x, y). Therefore player wats to make A as large as possible while player 2 wats to make A as small as possible. Example. The game played o [, ] [, ] with payoffs give by A(x, y) = 2x 4xy + y 2 is a zero sum iterval game. If player selects x = 2 ad player 2 selects, the player 2 would give player a payoff of, ) =. A( 2 A solutio to a iterval game cosists of three thigs: a optimal strategy for player, a optimal strategy for player 2, ad the value of the game (the value of the game is the expected payoff whe both players employ optimal strategies). However, ot every iterval game has a solutio! Example 2. The game who ca select the biggest umber? is a iterval game played o R R with payoff A(x, y) = if x > y ad A(x, y) = if x y. It has o optimal strategy sice there is o biggest real umber. Fortuately, some iterval games do have solutios. If I ad I 2 are both of the form [a, b] for a, b R ad if the payoff fuctio A(x, y) is cotiuous, the a solutio was proved to exist by Ville i 938. This was doe by approximatig iterval games with large matrix games. Eve whe a solutio is kow to exist, there are o kow efficiet techiques for fidig it aalytically. Fidig a geeral method for solvig iterval games without resortig to approximatios usig matrix games remais a ope problem i mathematical game theory.. SADDLE POINTS How ca we determie if a iterval game has pure strategy solutios for both players? I other words, uder which coditios are there umbers x I ad y I 2 such that player always selects x, player 2 always selects y, ad the value of the game is A(x, y )? This situatio is aalogous to the discrete problem of determiig which matrix games have saddle poits. Suppose x ad y are optimal strategies for the first ad secod players ad suppose that it is possible to take two partial derivatives of A(x, y). Sice player 2 always plays y, the umber x must maximize A(x, y ) for x I. Rememberig calculus, this meas that A x (x, y ) = ad that A xx (x, y ) <. Similarly, sice player always plays x, the umber y must miimize A(x, y) for y I 2. This meas that A y (x, y ) = ad that A yy (x, y ) >. This gives us a way to check if A(x, y) has a saddle poit solutio: () First, fid a poit (x, y ) with A x (x, y ) = A y (x, y ) =. (2) The, check to see if A xx (x, y ) < ad A yy (x, y ) >. If such a poit exists ad is foud withi I I 2, the player should always select x, player 2 should always select y, ad the value of the game is A(x, y ). Example 3. Cosider the game o [, ] [, ] with payoffs give by A(x, y) = 2x 2 + 2x 3xy + y Lookig for a saddle poit, the solutio to = A x (x, y) = 4x + 2 3y = A y (x, y) = 3x + 2y
2 2 ANTHONY MENDES is the poit ( 7 4, 7 6 ). We otice that A xx( 7 4, 7 6 ) = 4 < ad A yy( 7 4, 7 6 ) = 2 >, so we have foud the optimal strategies for player ad player 2. The value of the game is A( 7 4, 7 6 ) = The poit ( 7, 7 6 ) is a maximum whe x is allowed to vary ad a miimum if y is allowed to vary. I multivariate calculus, a poit satisfyig coditios ad 2 above is also called a saddle poit. However, calculus saddle poits may ot be game theory saddle poits. A calculus saddle poit is a poit which is a maximum whe lookig i oe directio ad a miimum i aother directio. A calculus saddle poit could, say, miimize A(x, y ) ad maximize A(x, y) istead of maximizig A(x, y ) ad miimizig A(x, y). Example 4. Cosider the game o [, ] [, ] with payoff fuctio A(x, y) = 4x 2 + 4y 2 + 2x + 2y 2xy. Lookig for a saddle poit, the solutio to = A x (x, y) = 8x + 2 2y = A y (x, y) = 8y + 2 2x is x = y = 2. This is ot a game theory saddle poit because A xx( 2, 2 ) = 8 <. However, it is a calculus saddle poit. It turs out that the solutio for this game is this: player should select with probability 2 ad with probability 2, player 2 should always select 2, ad the value of the game is. To verify that this is ideed a solutio, we eed to check that x s strategy maximizes A(x, 2 ) ad that y s strategy miimizes the expected payoff whe x employs his strategy. Ideed, if the secod player selects 2, the A(x, 2 ) = 4(x 2 )2 is maximized o [, ] at either or. If the first player selects with probability 2 ad with probability 2, the 2 A(, y) + 2 A(, y) = 4y2 4y + 2 is miimized at y = 2. This verifies that we have reached a equilibrium poit ad thus foud a solutio to the game. 2. CUMULATIVE DISTRIBUTION FUNCTIONS AND RIEMANN-STIELTJES INTEGRATION Solutios to iterval games ivolve cumulative distributio fuctios. Let X be a radom variable. The cumulative distributio fuctio for X is defied to be the fuctio F : R [, ] such that F(x) = (the probability that the radom variable X is x). Immediately from the defiitio, we ca see that F(x) is odecreasig, the probability that the radom variable X lies i (a, b] is F(b) F(a). lim F(x) =, lim F(x) =, ad x x Example 5. Suppose that we decide to select x = 3 with probability 2 ad x = 2 3 with probability 2 whe playig a iterval game. The distributio describig this solutio is give by if x (, 3 ) F(x) = 2 if x [ 3, 2 3 ) if x [ 2 3, ). Example 6. Some distributio fuctios are cotiuous. We may decide to play a game o [, ] [, ] accordig to the distributio F(x) give by if x (, ] F(x) = x if x (, ] if x (, ). Here, the probability that x ( 3, 2 3 ] is selected is equal to F( 2 3 ) F( 3 ) = 3. For cotiuous distributios F, the aswer to the questio what is the probability that you will play x (a, b]? is F(b) F(a). However, although it may seem strage at first, the aswer to the questio what is the probability that you will play x = a? is always. To see why this is true, cosider selectig a umber i [, ] by radomly selectig each digit i the umber s decimal expasio. What is the probability of selectig /3 =.3333 usig the distributio i example 6? The probability of radomly selectig the first 3 i this decimal expasio is /. The probability of radomly selectig both the first ad secod 3 s
3 INTERVAL GAMES 3 is / 2. More geerally, the probability of radomly selectig the first k 3 s is / k. Sice / k as k, the probability of gettig exactly /3 is ideed. Suppose that whe playig a iterval game with payoff A(x, y), player uses F(x) while player 2 always selects y. I order to check that the strategy for player is optimal, we eed to check that F maximizes the expected payoff whe y is used by player 2. To do this, we first eed to uderstad how to calculate such a expected payoff. To fid the expected payoff, break I ito subitervals (x, x ], (x, x 2 ],..., (x, x ] just like whe defiig the Riema itegral. The expected payoff o iterval from (x i, x i+ ] is approximately (the payoff at x i, y )(the probability that x is i (x i, x i+ ]) = A(x i, y )(F(x i+ ) F(x i )). Addig the probabilities from all of these itervals together ad takig the limit as, the expected payoff is () lim A(x i, y )(F(x i+ ) F(x i )). This is kow as a Riema Stieltjes itegral ad it will be deoted by A(x, y ) df(x). The Riema Stieltjes itegral i () is much the defiitio of the regular Riema itegral of A(x, y ) o the iterval I ; that is, () is much like I A(x, y ) dx = lim A(x i, y )(x i+ x i ). The oly differece betwee the regular Riema itegral ad the Riema Stieltjes itegral is that the (x i+ x i ) has bee replaced with (F(x i+ ) F(x i )). Theorem. Suppose that F (x) exists ad is bouded I. The A(x, y) df(x) = A(x, y)f (x) dx. I Proof. We have A(x, y) df(x) = lim = lim = lim A(x i, y)(f(x i+ ) F(x i )) A(x i, y) F(x i+) F(x i ) x i+ x i (x i+ x i ) A(x i, y)f (x )(x i+ x i ) where the last lie used the mea value theorem to tur (F(x i+ ) F(x i ))/(x i+ x i ) ito F (x ) for some x (x i+, x i ). This last lie is the desired Riema itegral. Now we are ready to formulate our defiitio of solutio. Suppose that whe playig the iterval game A(x, y), player uses the cumulative distributio fuctio F(x) ad player 2 uses the cumulative distributio fuctio G(y). This is a solutio provided there is a v R such that A(x, y ) df(x) v ad A(x, y) dg(y) v for all x I ad y I 2. I other words, player ca guaratee a expected payoff of at least v usig F ad player 2 ca guaratee a expected payoff of at most v usig G. These cumulative distributio fuctios F ad G are called optimal. If I is ot bouded, we ca chage the first iterval to be (, x ) ad the last iterval to be (x, ) as eeded.
4 4 ANTHONY MENDES Example 7. Let A(x, y) = be a game o [, ) [, ). Take if x <, F(x) = e x if x, x if x < y y if x y ad G(y) = if y <, if y. We claim that a solutio is give by F(x), G(y) ad the value of the game is. To verify that F(x) is optimal for player, we calculate A(x, y) df(x) = ( x)e x dx + ( y)e x dx =. y Therefore, the least that player 2 ca achieve whe player uses F(x) is. As for player 2, the Riema- Stieltjes itegral A(x, y) dg(y) is really just A(x, ). Sice A(x, ) =, this meas that the most that player ca achieve whe player 2 uses G(x) is. We have verified that we ideed have a solutio. 3. A WAY TO SOLVE SOME ZERO SUM SYMMETRIC GAMES Here is a problem similar to exercise 3: Two me start ruig towards each other with loaded pistols draw startig at time t =. Each pistol has oe shot. At time t =, uless oe ma is already dead, the two me will be stadig face to face. Each ma has a probability of t of killig his oppoet if he waits t secods to fire. The duel is silet, meaig that each ma does ot kow if their oppoet has already fired (uless they are hit by their oppoet s bullet). Whe should each ma fire his pistol? Let A(x, y) be the fuctio givig the payoffs for this game o the square. If both players miss or if both players fire ad hit each other at the same istat, the payoff will be. Otherwise, we will assig a payoff for the successful participat. Suppose player selects x ad player 2 selects y with x < y. Player ca wi istatly with a successful shot; this happes with probability x. If player misses, which happes with probability x, the player 2 will wi with probability y. Therefore, if x < y, A(x, y) = x + ( x)y( ). Similar reasoig whe y < x ca be employed to fid that x y( x) if x < y A(x, y) = if x = y y + x( y) if y < x. This game is symmetric (meaig that the game is the same for either player) ad so the value should be. There should also be oe strategy give by a distributio F which is optimal for both players. Our goal is to fid this fuctio F. We will assume F is costat except o some iterval (a, b], F (x) exists o (a, b), ad F (x) > for all x (a, b). Hopefully we will be able to fid a F uder these assumptios; if ot, the maybe these assumptios are too strog. Suppose the expected payoff whe player uses F is a costat v regardless of the strategy employed by player 2. This would mea that the value of the game is at least v. But, sice this game is symmetric, player 2 ca also use F to guaratee a value of at most v; implyig that F is optimal for both players ad the value of the game is ideed v. So, to solve this game, we will search for a strategy F which makes the expected payoff whe player uses F ad player 2 uses ay y [, ] a costat. I symbols, this says = v = = A(x, y)f (x) dx (x y( x))f (x) dx + ( y + x( y))f (x) dx. y Rewritig this equatio, ad usig F (x) dx = (this is true for ay cumulative distributio fuctio), we fid = xf (x) dx + y xf (x) dx + y xf (x) dx y.
5 INTERVAL GAMES 5 Takig / y o both sides of this equatio ad simplifyig, we have = 2y 2 F (y) + xf (x) dx + xf (x) dx. Takig / y agai ad simplifyig, we fially arrive at = 6yF (y) + 2y 2 F (y). Studets who have take a elemetary course i differetial equatios may recogize that we have foud a Cauchy-Euler differetial equatio. The solutios are fuctios of the form F(y) = y m for some umber m. This meas F (y) = my m ad F (y) = m(m )y m 2. Pluggig these fuctios ito the differetial equatio ad solvig for m, = 6y(my m ) + 2y 2 (m(m )y m 2 ) = (2m 2 + 4m)y m, ad so 2m 2 + 4m =. This meas m = or m = 2 ad therefore, o (a, b], F(y) = C + C 2 y 2 for some costats C, C 2. At this poit, we kow that if x a F(x) = C + C 2 x 2 if x (a, b] if x > b. To fiish, we eed to fid a, b, C, ad C 2. Now, E(F, b) = b a b a = E(F, ) (x b( x))f (x) dx (x ( x))f (x) dx where the middle iequality betwee itegrals is because b. We kow E(F, b) = ad E(F, ), the iequalities i the above calculatio must be equalities. This implies b =. Sice E(F, ) =, = E(F, ) = = a a (x ( x))f (x) dx (2x )( 2C 2 x 3 ) dx = ( 2C ) 3a2 + 4a 2a 2. Therefore, 3a 2 + 4a =. This gives a = 3 or a =. Sice the solutio a = does ot make sese here, we have foud that a = 3. We ow have that F( 3 ) = ad F() =. Therefore, = F() F(/3) = /3 F (x) dx = /3 2C 2 x 3 dx = 8C 2 ad we have foud C 2 = 8. Fially, usig = F() = C 8, we fid that C = 9 8. Puttig everythig together, the solutio to the duel problem that both players should fire their weapo accordig to the distributio F give by if x 3 F(x) = x 2 if x ( 3, ] if x >. So, for example, uless player is dead, he will shoot before time 2 with probability F( 2 ) = 5 8 =.625. Here what we just did i order to solve the duel game: () Foud a explicit expressio for A(x, y).
6 6 ANTHONY MENDES (2) Argued that if the game had a solutio, the it has a value of. (3) Assumed there was a commo cumulative distributio fuctio F(x) which was optimal for both players such that F (x) exists ad is ozero oly o some iterval (a, b]. (4) Foud a equatio formed from the expected payoff whe player does F ad player 2 does y. (5) Differetiated (twice, i this example) the equatio i foud i 4. (6) Solved the differetial equatio to fid the form of F(x). (7) Solved for the costats a, b ad those comig from the solutio of the differetial equatio usig properties of cumulative distributio fuctios. This approach works for other symmetric games, like those foud i exercises 9 ad. 4. NONZERO SUM GAMES A ozero sum iterval game has a payoff fuctio A : I I 2 R R. The first coordiate of A(x, y) is iterpreted as the payoff to player ad the secod coordiate is the payoff two player 2. As usual, both players wat to maximize their respective payoffs. The cocept of a solutio for a ozero sum game is difficult to defie. However, it still makes sese to discuss equilibria. Suppose that player uses the cumulative distributio fuctio F ad player 2 uses G. If the expected payoff to player 2 whe player uses F is costat ad the expected payoff to player whe player 2 uses G is also (a possibly differet) costat, the F ad G give a equilibrium poit. If A is a symmetric ozero sum game, the there may be oe strategy F which ca be used by both players to produce a equilibrium poit. If the expected payoff to player whe this F is used agaist ˆF is strictly greater tha the expected payoff whe ˆF is used agaist itself for all strategies ˆF = F, the F is called a evolutioarily stable strategy (ESS). Evolutioarily stable strategies have bee used to explai the behavior of orgaisms which evolve uder the forces of atural selectio sice, whe adopted by a populatio of players each playig two perso games, a ESS caot be ivaded by ay alterative strategy. Accordig to the defiitio, checkig to see if a equilibria strategy F is a ESS requires calculatig the expected payoffs of F versus ˆF ad ˆF versus ˆF for all strategies ˆF. This is dautig sice there are a ifiite umber of strategies ˆF to check! Luckily the ext theorem, which we state without proof, provides a shortcut which cuts dow the work cosiderably. Theorem 2. Suppose A is a symmetric ozero sum game ad F, whe used by both player ad player 2, is a equilibrium poit. The F is also a ESS if the expected payoff whe F is used agaist y is strictly greater tha the expected payoff whe y is used agaist itself for all poits y. Example 8 (The war of attritio). Startig at time t =, two players try to itimidate their oppoet util oe retreats, leavig a reward of utility r behid. Both players icur a cost depedig o the legth of the stadoff. The payoff fuctio defied o [, ) [, ) is give by ( x, r x) if x < y A(x, y) = (r/2 x, r/2 x) if x = y (r y, y) if x > y. There is a pure-strategy asymmetric equilibria for this game: player selects t = r ad player 2 selects t =. If player uses t = r, the player 2 would have o icetive to deviate from the strategy of playig t = sice ( r, ) if r < y A(r, y) = ( r/2, r/2) if r = y (r y, y) if r > y ever provides player 2 a opportuity to ear more tha. A similar calculatio shows that player would have o reaso to chage from the strategy r whe player 2 uses y =. This equilibria caot be a ESS because both players are ot usig the same strategy. I search for a ESS, let us suppose there is a differetiable distributio fuctio F which, whe used by both players, is a equilibrium poit. Whe player uses F, the expected payoff to player 2 is some
7 INTERVAL GAMES 7 costat. Therefore, costat = = Takig / y o both sides of this equatio ad simplifyig, (r x)f (x) dx + ( y)f (x) dx y (r x)f (x) dx + ( y)( F(y)). = rf (y) + F(y). This is relatively easy differetial equatio to solve sice it is a first order liear differetial equatio. It may be solved usig the itegratig factor or by separatig the variables ad itegratig. Whe this is doe, we fid F(y) = + Ke y/r where K is a costat. Sice F() =, the costat K =. Therefore, the desired fuctio F is if x < F(x) = e x/r if x. Is this fuctio F a ESS? Take y [, ). The expected payoff to player whe player uses F ad player 2 uses y is ( ) ( ) ( x) r e x/r dx + (r y) y r e x/r dx = 2re y/r r. The expected payoff to player whe both oppoets use y is /2 y. Usig Theorem 2, we eed to see whether or ot 2re y/r r > /2 y for all y. To do this, we cosider the fuctio 2re y/r r (/2 y). Differetiatig this fuctio ad settig equal to, the miimum of this fuctio occurs at r l 2. The actual miimum of this fuctio is /2 + r l 2. Whe this quatity is positive, F is a ESS; otherwise F is ot a ESS. Therefore, F is a ESS provided r > /(2 l 2).72; otherwise F is ot a ESS. 5. EXERCISES. Take a, b R. Show that the game o R R with payoffs give by A(x, y) = (x a) 2 (y b) 2 has a o solutio. 2. Take a, b R. Show that the game o R R with payoffs give by A(x, y) = (y b) 2 (x a) 2 has a saddle poit solutio. 3. Solve the oisy duel problem: Startig at t =, two me start ruig towards each other with loaded pistols. Each ma has oe bullet to fire at their oppoet. At time t =, uless oe ma is dead by the, the two me will be stadig face to face. Each ma has a probability of t of killig his oppoet if he waits t secods to fire. The duel is oisy, meaig that each ma kows if their oppoet has already fired ad missed. Whe should each ma fire his pistol? 4. Cosider a zero sum iterval game with payoff fuctio A(x, y) which value v. Take a, b R such that a. Show that a optimal strategy for A(x, y) is still optimal for aa(x, y) + b ad the value of aa(x, y) + b is av + b. 5. Show the set of optimal strategies is covex; that is, show that if F (x) ad F 2 (x) are two optimal strategies for player i a iterval game, the so is λf (x) + ( λ)f 2 (x) for all λ [, ]. 6. Let A(x, y) be a iterval game o I I for some iterval I such that there is oe cumulative distributio fuctio F which is optimal for both players. Solve the iterval game o I I with payoffs give by A(x, y) A(y, x). 7. Two players idepedetly select umbers i (, ). The player who selected the smaller umber, say t, pays e 2t to his oppoet (there is o payoff if they both select the same umber). By usig the strategy F(x) = e x for x >, how much ca a player guaratee for himself? Does this game have a value?
8 8 ANTHONY MENDES 8. Player selects x [, 3] while player 2 selects y [, 2]. The the secod player pays the first player $ if x (y, y + ) ad $ otherwise. Show that a optimal strategy for the first player is to use the cumulative distributio fuctio if x <, F(x) = x/3 if x [, 3], ad if x > 3 ad a optimal strategy for the secod player is to select each of the umbers, ad 2 with probability /3. What is the value of the game? 9. Two people try to guess a radom umber foud usig the cumulative distributio fuctio if x <, F(x) = x if x [, ], if x >. The perso comig closest without guessig too high wis. Solve. As a extra challege, solve whe F(x) is replaced with a arbitrary differetiable cumulative distributio fuctio.. A beautiful woma will arrive at the airport at some time i the iterval [, ]. The probability that she will arrive at or before time t [, ] is t. Two hadsome suitors will each visit the airport lookig for the woma i order to give her a ride home. If a suitor arrives whe the woma is ot there, he will immediately leave uder the assumptio that she has already bee picked up. The suitor who successfully picks up the woma wis from the other suitor (if there is a tie or both suitors arrive before the woma, both suitors receive ). Whe should each suitor arrive? MATHEMATICS DEPARTMENT, CALIFORNIA POLYTECHNIC STATE UNIVERSITY, SAN LUIS OBISPO, CALIFORNIA address: aamedes@calpoly.edu
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