CS286r Assgn One Answer Key 1 Game theory 1.1 1.1.1 Let off-equlbrum strateges also be that people contnue to play n Nash equlbrum. Devatng from any Nash equlbrum s a weakly domnated strategy. That s, any alternatve strategy s has a lower payoff n expectaton than the Nash equlbrum strategy n each stage game. Ths mples that any devaton n the supergame s also weakly domnated by playng a Nash equlbrum n each round. Snce any devaton s weakly domnated the supergame s n subgame perfect equlbrum. 1.1.2 No, these are not the only subgame perfect equlbra of supergames. Grm trgger strateges n the repeated prsoner s dlemma, for example, are subgame perfect equlbra but a Nash equlbrum s not played n each stage game. 1.2 1.2.1 Along the same lnes as the no-trade theorem, Bayesan agents cannot smultaneously both agree they wll proft n a zero-sum game, mplyng both wll ext. Wthout loss of generalty we restrct attenton to two players wth a fnte set of sgnals each. Let X be a jont dstrbuton over the sgnals and π some proposed strategy profle. We can then construct a 2-dmensonal matrx of the expected score for each player gven both sgnals, and each entry n the matrx can be fully descrbed by the expected payoff to the row player. Ths produces Left Rght Top 3-5 Bottom 4 - Each player s expected score for playng under π gven ther sgnal s then a matrx lke the sum of values n each row or the negatve sum of each column (assumng, wthout loss of generalty, that all sgnals are equally lkely. If not we can fnd the 1
greatest common dvsor and break sgnals nto equvalent parts). If the sum of a row s negatve, or sum of a column s postve, then the row (resp. column) player wll ext, so those rows and columns must be removed. Ths reduces our orgnal queston to: can there exst a real-valued matrx where every row sums to a strctly postve number and every column sums to a strctly negatve one? One last smplfyng assumpton gves us an easy answer to ths queston. Assume that the postve and negatve values both use the same real, e.g. every entry n the matrx s ether r or r. Ths s, agan, wthout loss of generalty because we can fnd a gcd and expand each row (also duplcatng and extendng columns) and preserve all sums whle growng the matrx. Then our queston becomes: can there exst a matrx wth more postve terms n each row and more negatve terms n each column? But ths s easly seen to be mpossble snce then most terms would need to be postve and most terms would need to be negatve! Thus we conclude such a matrx cannot exst, and the expected score for each player must be zero (or negatve, but then they would ext). From the technque we can deduce the extenson readly to more than two players. 1.2.2 Followng the no-trade lne of thought a market maker must subsdze a predcton market (make t a constant-sum game) to encourage ratonal agents to partcpate. Wthout a subsdy the game would be zero-sum and all agents mght ext. 2 Buldng a scorng rule 2.1 2.1.1 It s strctly convex. Consder the functon g(p + λq), then t has dervatves dg dλ = q log(p + λq ) d 2 g dλ 2 = 1 q 2 /(p + λq ) log 2 The second dervatve s strctly postve, mplyng the functon s strctly convex. 2.1.2 s(p, ) = g(p) g (p) p + g (p) = p log p p log p + log p = log p 2
2.1.3 The logarthmc scorng rule. 2.2 2.2.1 It s derved from a strctly convex functon. So followng G + R t s strctly convex. 2.2.2 2.3 2.3.1 q log p When an expert s reportng honestly (q == p) they are the same functons. Varous answers are acceptable. The most smple may be that the net score s undefned wthout a pror predcton. 2.3.2 The market maker s worst-case loss s max p log p log p 0 where p 0 s the market s ntal predcton and the logarthmc scorng rule s used. More abstractly we would replace the logarthmc functon n the above wth the scorng rule used. The worst-case loss s realzed when the last expert predcts the realzed outcome perfectly, snce experts always score more for more accurate predctons when usng a strctly proper scorng rule. To mnmmze the worst-case loss we solve sup p nf log p whch (straghtforwardly) has a soluton at the unform vector. Equvalently, the market maker mnmzes ts worst-case loss (when usng the logarthmc scorng rule) by settng ts ntal dstrbuton to an entropy maxmzng one. 2.3.3 q (log p log p ) 3
2.3.4 The score of prevous predctons n a predcton market s not affected by your own predcton. Snce they are constant and solvng an arg max expresson s ndependent of addng or subtractng constant terms, a strctly proper scorng rule maxmzes both a score and net score. 2.4 2.4.1 2.4.2 C(Y ) C(X) = b ln(e Y1/b + e Y2/b ) ln(e X1/b + e X2/b ) The mpled probablty s the prce and mutats mutand for p 2. 2.4.3 p 1 = e X1/b /(e X1/b + e X2/b ) The b parameter controls the rate of change of the market or equvalently ts lqudty. A hgher b means that trades wll have a smaller effect on prces. 2.4.4 Prces are ntrade are n a 1:1 correspondence wth the market s belef of the lkelhood. If the prces s below your beleved probablty you wll buy contracts and f t s above your belefs you wll sell t. 3 Scorng Experts The followng table constructed used Bayes s theorem wll be useful. Far F 11 1 C 3 49 FF 65 FC CF 1 CC 5 Cloudy 4 11 2 3 16 65 8 8 4 5 3.0.5 See the above table. 4
3.0.6 Wth a flat fee an expert has no mmedate ncentve to be accurate snce t s pad the same free regardless of whether t s accurate or not. 3.0. Let c be the cost of acqurng addtonal nformaton. Assume the frst sgnal s F, then the expected score maxmzng predcton s (/11, 4/11). If the true state of the world s far (whch t s 11 of the tme) and f we purchase an addtonal sgnal we get F F /10 of the tme and score log 49/65, otherwse we get F C and score log /. Otherwse, 4/11 of the tme the world s cloudy and 4/10 of the tme we get log 16/65, otherwse we get log 8/. So we compare 11 ( 49 log 10 65 + 3 10 log ) + 4 11 ( 4 16 log 10 65 + 6 10 log 8 ) 11 log 11 + 4 11 log 4 11 ) + c and puttng ths nto Mathematca gves us a value of c.043. Usng the same reasonng for the case where we get C we fnd that c.0408. So f c s less than (roughly).0408 the expert wll always purchase addtonal nformaton. 3.0.8 Gven a value for c as our cost, then we take the least c from the prevous queston (.0408) and solve b = c/c. 3.0.9 Incentves are not affected by the addtve term a snce the change n the functon s ndependent of the predcton qualty. It may, however, be useful to have a postve a to attract a partcpant. Especally snce the logarthmc scorng rule s always negatve, so extng s always a preferred strategy f a = 0. 4 Some research usng scorng rules 4.0.10 Followng the paper where n s the number of outcomes. 4.0.11 p = π u (m ) j π ju (m j ) = π j π j = 1 n The same snce the probablty dstrbuton π and the dervatve of the utlty functon are nvarant. 5
4.0.12 Boundng u below practcally means that the market maker wll not accept trades that mght cause t to lose more than 5 dollars. 4.0.13 Ths market maker has bounded loss, so n that sense t s better. It achves ths by denyng some trades, however. 4.0.14 Let y be the shares outstandng n the market, and x the shares outstandng on outcome x. Then we have that must hold for a trade to occur. (y + 20)/n x + 20 5 6