Colonel Blotto with Imperfect Targeting

Transcription

1 Departent of Econoics Working Paper Colonel Blotto with Iperfect Targeting Deborah Fletcher Miai University Steven Slutsky University of Florida Deceber 2009 Working Paper #

2 Colonel Blotto with Iperfect Targeting* Deborah Fletcher a, **, Steven Slutsky b a Departent of Econoics, Miai University, 208 Laws Hall, Oxford, OH b Departent of Econoics, University of Florida, Matherly Hall, Gainesville, FL Deceber 2009 Abstract: Colonel Blotto gaes have been widely applied in a variety of contests where the players allocate resources across a nuber of battlefields and the winner in each battlefield is the one with ore resources there. One drawback of this standard odel is the assuption that players can perfectly target their efforts toward different battlefields. In any situations, players can only iperfectly target different battlefields, with an allocation affecting ore than one battlefield. We develop an extension of Colonel Blotto that incorporates this type of interrelation. The equilibria here generally utilize ixed strategies that tend to be asyetric across battlefields. That is, at least one player will have zero probability of dividing resources equally across battlefields, even if the characteristics of the battlefields are siilar. We use advertising data fro the 2002 races for governor and US senate to test the application of the Blotto odel to political contests. Although the Colonel Blotto odel has been used extensively to odel political capaigns, we find that candidates advertising allocations across arkets are far too syetric to fit the odel. JEL code: C72 Keywords: Colonel Blotto; ixed strategies **Corresponding author. Eail addresses: fletchd@uohio.edu, steven.slutsky@cba.ufl.edu

3 (I) Introduction A variety of contests exist in which the opponents copete by expending resources. In any gaes of this type, the players benefit by allocating resources differently fro their opponents; the gae is thus one of strategic allocative isatch. 1 Proinent exaples of this are ilitary contests and political capaigns. One approach to odeling such situations is the classic Colonel Blotto gae. 2 In this odel, the winner is deterined by which player puts in ore resources, that is gets there first with the ost. 3 This akes the payoffs very discontinuous in the actions of the players. In the siplest version of Colonel Blotto, the players have equal resources that they ust allocate to different battlefields of equal size. On any battlefield, the winner is the player with the ost resources there. The goal of the players is to axiize the expected nuber of battlefields that they win. Extensions of this odel allow for the players to have unequal resources, for battlefields to be of different sizes, and for one of the players to have an advantage on soe battlefields so as to be able to win the with fewer resources (perhaps, having gotten there first, within liits, the player does not need the ost). 4, 5 1 See Golan and Page (2009). 2 Borel (1921) first proposed this gae, with further analysis in Borel and Ville (1938). Modern analysis began with Tukey (1949), with follow-ups by Blackett (1954, 1958) and Bellan (1969). Only recently has a general solution to the continuous gae been provided (Roberson (2006)); see also Weinstein (2005). 3 In response to a question about who would win a battle, Gen. Nathan Bedford Forest of the Confederate Ary replied, The one who gets there first with the ost en. This quotation is often used in discussions of strategy in a variety of contexts. 4 Kovenock and Roberson (2008) consider a odel of two-party copetition where parties attract voters by offering redistributive policies when voters have heterogeneous loyalties to the different parties. There is a foral overlap between getting votes through redistributive policies and attracting the through capaign expenditures, as analyzed here. 5 Colonel Blotto odels of advertising go back to Friedan (1958). Colonel Blotto odels of elections go back at least to Sankoff and Mellos (1972). Recently, Szentes and Rosenthal (2003) study a type of all-pay 2

4 Although the Colonel Blotto gae can be straightforward to specify, equilibriu behavior can be coplicated. Pure strategy equilibria exist only in special cases, such as when only two battlefields exist or when one battlefield is uch larger than all the others cobined. In the latter case, the pure strategy equilibriu has both players spending all resources in the larger battlefield. Pure strategy equilibria also exist if one player has such an overwheling resource advantage that it can allocate enough resources to the battlefields to win the all, regardless of the other player s allocation. Typically, when there are ore than two battlefields, only ixed strategy equilibria exist, and can be quite coplex. Consider a player axiizing over n battlefields. The arginal distributions of the ixed strategy give the probability of allocating different aounts to any battlefield, while the joint distributions give the probability of allocating different n-tuples across the battlefields. Necessary and sufficient conditions on the arginal distributions for a ixed strategy equilibriu have been specified (see Roberson (2006)). Consistent with any set of arginal distributions, ultiple joint distributions can exist. Characterizing the equilibriu joint distributions has been ore difficult. Exaples of these go back to Borel and Ville (1938) with other possibilities shown only recently by Roberson (2006) and Weinstein (2005). The equilibriu joint distributions can vary widely in their nature. Soe are asyetric and have supports that are of lower diension than the n-1 diensional space of allocations. There are also equilibria with full support such as the Hex equilibriu given by Borel and Ville; see Weinstein for a discussion of this equilibriu. In an equilibriu with full support, any neighborhood in auction of which a Colonel Blotto odel of Electoral College copetition is an exaple. Laslier and Picard (2002) consider a odel of distributive politics that reduces to a Colonel Blotto gae. 3

5 the space of allocations, including any containing the point at which resources are allocated equally to all battlefields, has positive easure. One assuption in all these variants is that players can perfectly target their resources. That is, resources can be allocated to a single battlefield with no spillover effects to other battlefields. In any contexts, however, spillovers do exist. For exaple, consider advertising capaigns by firs or political candidates. A battlefield ight be considered to be soe group defined by gender, ethnicity, age, socioeconoic status, geographic location, or preferences. In a perfect world for the players, they would be able to target expenditures in the for of ads with different essages very narrowly aied at each deographic group, with only that group seeing the ad designed for it. While firs or candidates do try to carry out such targeting through such actions as the placeent of different ads on specific radio or television progras, perfect targeting rarely is possible. Ads that are aied at one group will often be observed by ebers of other groups. 6 Fletcher and Slutsky (2009) develop a structure that allows considering such iperfect targeting in the probabilistic voting context. Their structure has two edia arkets where ads can be purchased. In each arket, the ads are viewed by three groups of voters: those initially supporting each candidate and those indifferent between the. The partisan types are assued to have an intensity of preference toward their preferred candidates. Ads ove this intensity in favor of the candidate running the ad. Under 6 Golan and Page (2009) also offer extensions of the Colonel Blotto gae that involve externalities across battlefields. However, their externalities are in the payoffs, with players valuing cobinations of battlefields beyond siply suing the values of each battlefield. They do not consider having the outcoes on ultiple battlefields depend on the sae expenditures as we do here. 4

6 probabilistic voting, the support each candidate receives fro a type varies continuously with its post-capaign intensity. In this paper, we adapt this structure to replace the probabilistic voting assuption so that, as in standard Blotto, all ebers of each group in a arket vote for the candidate the group prefers after all ads have been shown, no atter how sall the group s preference for that candidate. In addition, we generalize its applicability to situations beyond the political context. 7 We consider a situation in which ultiple battlefields are grouped together because they cannot be targeted separately, and the players allocate resources to each grouping. For convenience, we will call a grouping a battlefield and each coponent of the grouping a sub-battlefield. We will analyze allocations across two battlefields, each of which has three sub-battlefields. 8 The two battlefields overall ay be of unequal iportance and the different sub-battlefields can also vary in iportance. One of the players ay have an advantage on soe subbattlefield so can win there, even though she allocates fewer resources to that subbattlefield than the other player. The effectiveness of allocations can differ across battlefields or even across sub-battlefields. The effectiveness function is, however, the sae for both players. 9 Unlike the typical Colonel Blotto gae or the odel in Fletcher and Slutsky (2009), we also assue that allocations can have diinishing rather than constant arginal effectiveness. 7 Iperfect targeting has been analyzed in soe contexts, such as developent econoics (see Bibi and Duclos (2007)). However, it has not been explored in the contest literature. 8 While it would be interesting to analyze the case where there is a continuu of sub-battlefields, such an approach would no longer be in the spirit of a Blotto odel. In a continuu odel, the aount a player wins would vary continuously with the resources each side expends, while in a Blotto odel, there are significant discontinuities in the aounts won as functions of the resources expended. 9 This generalization has an iportant iplication in the political context. Since the effectiveness function ay differ across arkets, it incorporates the possibility of ad price differences across arkets. Thus, the decision variable could be either the nuber of ads played in the arket or the total advertising expenditure. 5

7 Our results under iperfect targeting are not only consistent with the basic nature of standard Blotto, but strengthen those results. First, pure strategy equilibria are even less likely to exist with iperfect targeting. Unlike the standard results, pure strategy equilibria ay not exist even when there are only two battlefields and one is larger than the other. As follows fro Theore 1 and Lea 2 below, the neutral sub-battlefield in one battlefield ust be larger than the entire other battlefield for a pure strategy equilibriu to exist. Of course, then the equilibriu has all resources expended in the larger arket. Second, consider situations where pure strategy equilibria do not exist and the players have the sae aount of resources to allocate between battlefields. If this aount lies in soe range deterined by the aount of the advantages the players have in the non-neutral sub-battlefields, a necessary condition on equilibriu ixed strategies is that at least one player acts asyetrically. That is, as shown in Theore 2 below, there cannot be equilibria where the joint distributions for both players have full support with at least one player putting no probability weight on allocations in a neighborhood around equal allocation across the two battlefields. 10 This contrasts with the standard Blotto result, which has ixed strategy equilibria with realizations where candidates allocate equally across arkets. Thus, for this odified Blotto odel, whether a pure or ixed strategy exists, players will use very asyetric strategies across battlefields. Given this, it would be possible in principle to test whether a Blotto approach is consistent with behavior in any of the specific contexts to which the analysis ight apply. If Blotto were appropriate, we would expect to see asyetric allocations across the 10 Note that although iperfect targeting introduces an eleent of having ore than two battlefields, an iportant eleent of just two battlefields reains. Taking into account budget balance, each player has only a one-diensional allocation decision. 6

8 battlefields, even when they appear to be siilar. Thus, an ary would allocate troops asyetrically between two battlefronts even if the sub-battlefields in the were siilar in iportance and in the extent of the advantage to one ary over the other. Product or political advertiseents would tend to be placed ore heavily in one arket over another, even when the deand conditions in the arkets appear to be very siilar. Finally, we would expect to see significant variation in educational quality across school districts, even when the districts are alike in ost easurable ways. Whether the equilibriu is in pure or ixed strategies, players will use very asyetric strategies across battlefields. However, since the conditions for existence of pure strategy equilibria are so extree, the equilibriu will generally be in ixed strategies that are typically not directly observable. A sall literature exists that addresses the use of ixed strategies by players in gaes. This literature considers two questions: Do players eploy ixed strategies? If so, are they choosing the optial ixed strategies? The first question has been considered in several ways. Merolla, Munger, and Tofias (2005) note that when ixed strategies are used, a player would gain by altering the chosen action once the opponent s strategy is known. They argue that in the 2000 US presidential election, Gore used ixed strategies, since he would have gained fro reallocating resources had he known what allocations Bush was aking across states. To ake this arguent, they ust deterine the effect that reallocating expenditures would have on the outcoe in different states. They base this analysis on coparing outcoes to pre-election polls. Chiappori, Levitt, and Groseclose (2002) argue that in a gae in which players use ixed strategies and ake a nuber of oves, each ove is a rando 7

9 realization of the strategy. Thus, their choices should exhibit independence over tie. They show that goalies and kickers in soccer atches behave consistently with using ixed strategies during penalty kicks since their choices show no serial correlation. The second question, optiality of the ixed strategies, is considered by Chiappori, Levitt, and Groseclose (2002) in the soccer context and Walker and Wooders (2001) with respect to tennis serves. Both sets of authors copute what the optial ixtures would be, and both conclude that players choose strategy ixes close to the optial ones. Walker and Wooders find that ore experienced players choose ixed strategies closer to the optial ones but all players ix ore than would be optial. The approaches used above cannot be applied in any settings including political capaigns. Often there is only one realization of a candidate s strategy, not a series of observations as with tennis serves or soccer kicks. Strategy choices cannot be easily related to outcoes. For exaple, polling data is not as extensive for lower-level political races as it is for presidential elections. The necessary condition we derive gives a novel way of testing whether the Blotto fraework is appropriate in various contexts even if the candidates use ixed strategies by directly exaining the actions of players to see if they ever choose syetric actions across battlefields. With this approach, it is not necessary to ake epirical inferences about the effects of strategy choices on outcoes. For any of the applications, getting the appropriate data on players strategic choices would be difficult. For the case of political capaigns, however, detailed data exists on political advertising, one iportant strategic variable available to candidates. Recent data on political capaigns in the United States fro the Capaign Media Analysis Group (CMAG) contains extensive inforation on political advertiseents at 8

10 the individual ad level including content, length, tie, station of airing, and estiated cost. The data, sold to candidates and news organizations desiring real-tie inforation about capaign activity, are released for acadeic use after a lag of several years. The only data currently available for a non-presidential year are fro Despite soe shortcoings, that candidates purchase these data indicates that they contain useful inforation about strategic choices. 11 In general, the candidates see to use strategies which are not consistent with what the Blotto odel would predict, with advertising allocations uch ore equally divided across arkets. 12 This supports the conclusion that equal allocations ight be explained by a ore continuous gae of allocative isatch, such as a probabilistic voting odel as in Fletcher and Slutsky (2009) or a odel using a contest success function. 13 Of course, this evidence does not argue that the Blotto odel is not appropriate in any of the other settings to which the odel ight apply. The foral odel is described in Section II. Pure strategy equilibria are considered in Section III. Soe properties of ixed strategy equilibria are derived in Section IV. The epirical analysis related to political capaigns is given in Section V and conclusions are discussed in Section VI. All proofs are contained in an Appendix. 11 The data are only for broadcast and cable television in the 100 largest edia arkets, and do not include advertising on other edia such as print or radio. Soe aspects of the data are questionable, particularly the price coponent. See Goldstein and Freedan (2002) for a detailed discussion of the CMAG data and its liitations. 12 This situation differs fro the outcoe that is observed in Presidential elections, where candidates allocate asyetrically across states. In the forer, the agnitude of the win in a battlefield (arket) atters because it is the su of votes over all the arkets in a state that atters. In the latter, the agnitude of a player s win in a battlefield (state) is uniportant, since a player wins all the Electoral College votes of the state if he wins that state by the narrowest of argins. 13 Skaperdas (1996) axioatizes several contest success functions that have been widely used in the literature. 9

11 (II) The Model Consider a contest between two players A and B, each of who divides resources between two battlefields denoted and n. Each battlefield has three sub-battlefields denoted 1, 2 and 3. Sub-battlefield (i, j) denotes sub-battlefield i of battlefield j. Each sub-battlefield possesses two characteristics: iportance and advantage. Iportance ay be the size of the sub-battlefield, but could ore generally relate to its significance to the players. The iportance of sub-battlefields are denoted by i and n i, i = 1, 2, 3. The overall iportance of a battlefield is assued to be the su of the iportance of the sub-battlefields, with M = i and N = n i. The two players have the sae evaluation of the iportance of sub-battlefields. Advantage relates to the extent to which the characteristics of a sub-battlefield favor one of the players over the other. For convenience, advantages to A are treated as positive while advantages to B are treated as negative. Sub-battlefield 1 in each battlefield favors B with the size of the advantage denoted as - and - n, while sub-battlefields 3 favor A with the size of the advantage denoted as and n. Sub-battlefields 2 are neutral, with neither player having an advantage before the battle begins. The total resources available to the two players are R A and R B, which we assue are exogenously deterined. The players can allocate their resources across the two battlefields with x and x n denoting the allocations of player A and y and y n denoting the allocations of player B. Thus, the two players face resource constraints x + x n = R A and y + y n = R B (1) An allocation to a battlefield goes to all the sub-battlefields in that battlefield in the sae aount. Resources allocated to a battlefield by a player will shift the advantage toward 10

12 that player. The effectiveness of resources in shifting the advantage are given by the functions h ij in sub-battlefield i of battlefield j, where h 0,h 0,h (0) 0 (2) ij ij ij For exaple, in sub-battlefield (1, ), the post-battle advantage to player B is + h 1 (x ) h 1 (y ), while in sub-battlefield (3, n), the post-battle advantage to player A is n + h 3n (x n ) h 3n (y n ). Figure 1 shows the post-battle situation in the two battlefields. This specification generalizes the standard Colonel Blotto assuption that effectiveness is linear in expenditures. Here, we allow for the possibility of diinishing arginal effectiveness. In addition, specifying different functions for each sub-battlefield allows the effectiveness of allocations to differ across sub-battlefields. This could be due to differences between the sub-battlefields in aspects such as size and terrain. We continue to assue that the effectiveness functions are the sae for both players and are independent of the level of the other player s allocation. Any sub-battlefield is won by the player with the post-battle advantage. Let I denote the post-battle advantage to player A. Then the outcoe in any sub-battlefield is specified as: { 1 if > 0 V() = 1\2 if = 0 0 if < 0 (3) The overall payoff to Player A is Vˆ (x, x n, y, y n ) = 1 V[- + h 1 (x ) h 1 (y )] + 2 V[h 2 (x ) h 2 (y )] + 3 V[ + h 3 (x ) h 3 (y )] + n 1 V[- n + h 1n (x n ) h 1n (y n )] + n 2 V[h 2n (x n ) h 2n (y n )] + 11

13 n 3 V[ n + h 3n (x n ) h 3n (y n )] (4) The players siultaneously ake their allocation decisions, with Player A choosing x and x n to axiize Vˆ and player B choosing y and y n to iniize Vˆ, subject to the resource constraints (1). Consider the following settings that fit within the context of this odel. Military capaigns First, to extend the classic Colonel Blotto warfare odel to incorporate iperfect targeting, consider a ilitary capaign between ary A and ary B. Two battlefields, and n, are contested. Ary A allocates x soldiers to battlefield and x n soldiers to battlefield n, while ary B allocates y and y n soldiers across the battlefields. In each battlefield, the aries face off across a linear battlefront, with the terrain giving advantage to one ary or the other at different points along the battlefront. Subbattlefields 1 and 3 denote points along the front that are advantageous to ary B and ary A, respectively, while sub-battlefields 2 are neutral, favoring neither ary. For exaple, one location in battlefield ight have boulders on ary B s side of the front, giving that ary s soldiers convenient cover. The agnitude by which ary B gains fro this cover is represented by. The iportance paraeter 1 could be based on the size of this location or on its intrinsic value to the aries. Such points of advantage need not be geographically contiguous, so that it would be difficult for an ary to send soldiers only to neutral points or to those points where that ary has an advantage. Thus, each ary is unable to perfectly target its troops to areas with a given advantage. The 12

14 effectiveness of troops, represented by the h ij functions, could differ across subbattlefields because of differences in such factors as ease in counications or resupply. Advertising capaigns A second application of the odel is in advertising capaigns, either between two firs copeting for sales, or two political candidates copeting for votes. In the product advertising case, the firs do not copete by changing the characteristics of the products, but through advertising. Consuers differ in which product they prefer, with soe being indifferent. For exaple, a few years ago the University of Florida switched fro selling only Coke products to selling only Pepsi products on capus. Soe at the University were ecstatic about the change, others were outraged, and soe, like the authors, were copletely indifferent. The Coca-Cola Copany and PepsiCo do not attept to win consuers over by changing the forulas of Coke and Pepsi, but instead have advertising wars. As in Golan and Page (2009), the product advertising case is winner-takes-all if a store has shelf space for only one fir s product. 14 Another possibility is advertising to institutional users that will sign exclusive contracts to use one fir s good. Exaples of this would be hospitals buying all their cleaning supplies fro a single supplier, or schools buying all their ilk fro a single producer because of the influence of advertising. In the political advertising exaple, the two candidates have platfors that are fixed prior to the beginning of the capaigns. The capaign of each candidate is designed to alter voter preferences in favor of that candidate. Since no voter is likely to 14 If soe custoers use arket ranking as a signal of quality, then a discontinuous payoff would result siilar to the winner-takes-all setting. A relatively sall increase in advertising ight increase the fir s arket ranking, then induce a large increase in sales through the quality perception effect. 13

15 prefer one candidate s position over that of the other on every issue, ads ight be designed to change voter preferences in other ways. For exaple, candidates ight attept to change the saliency of different issues in the inds of the voters, or focus on non-issue factors such as character or copetence. 15 The battlefields in either advertising case would be edia arkets and n. In the political case, these are different edia arkets in the sae electoral district, such as two cities in the sae state for a Senate race. Subarkets 1 represent consuers with a preference for product B or partisans for candidate B, while subarkets 3 are the individuals preferring product or candidate A. Subarkets 2 are the neutral consuers or voters. Firs or candidates allocate advertising dollars x and y across arkets, but cannot perfectly target who will see a given ad within a arket. Since all sales and votes are equally iportant, 16 the iportance paraeters i and n i siply represent nubers of consuers or voters. j and j are the preference intensities of the partisans for B and A, respectively. The h ij are the advertising effectiveness functions in the various subarkets. Ads ight have different effectiveness in different edia arkets because of different television viewing habits or differences in ad prices between large and sall arkets. Additionally, individuals with initially strong preferences toward a product or candidate ight respond differently to ads than do initially neutral individuals. 15 In the political context, axiizing the probability of winning rather than expected plurality is considered preferable, but is often not used as it is significantly ore coplicated to analyze. A nuber of papers including Aranson, Hinich and Ordeshook (1974), Ledyard (1984), Snyder (1989), Duggan (2000) and Patty (2005) have considered the relation between outcoes under the two objectives in a spatial voting context. In soe, perhaps special, circustances, they are equivalent. It should be noted that expected plurality axiization is not only ore convenient but in soe circustances can be justified as ore realistic. Candidates with a low probability of winning ay desire to lose by as sall a argin as possible. 16 In the political setting, this would be true in a statewide or district-wide race. Obviously, all votes are not created equal in an Electoral College setting. 14

16 Quality copetition aong providers of education In the final exaple, two private school providers copete for students by varying the quality of education in two districts, and n. Provider A is a religious organization and provider B is a secular fir. Quality differences are achieved by A via allocations x and x n, and by B via y and y n. The providers wish to axiize the total nuber of pupils they serve because of econoies of scale arising fro such factors as reduced textbook prices for large orders or fixed adinistrative costs. Different types of parents have underlying preferences about sending their children to secular or religious schools, while soe parents are indifferent. These underlying preferences i and j could be overcoe with large enough quality differentials. Quality differentials are difficult to target toward pupils of one type and not another. 1, 3 and 2 are the parent types preferring secular or religious education and the neutrals, respectively. h ij, the effectiveness of dollars spent on quality, ight differ across counities because dollars buy different aounts of quality due to differences in land costs and prevailing wage rates, or due to differently sized physical plants. All of these factors would likely be correlated with counity size. The effectiveness functions ight also differ across parental types, as these ay have different sensitivities to changes in quality. The iportance paraeters would generally relate to the nubers of the different types in each counity but ight also incorporate desires by the schools to have students with certain skills (acadeic or athletic). Within the general structure specified above, we ake soe notational conventions and soe technical assuptions to assure that the analysis is interesting. 15

17 Without loss of generality, we can renae the players so that A has at least as uch resources as B (i.e. R A R B ). In addition, we can renae the battlefields so that n is at least as iportant as (N M). If the player advantages in sub-battlefields 1 and 3 were so large as to be insurountable by any allocation of resources by the opponent not favored in the, then the gae would reduce to a single battlefield, and the extension would not be interesting. Thus, we assue that all sub-battlefields are in play. That is, each player has sufficient resources to potentially win those sub-battlefields in which the other player has an advantage: 17 j < h 1j (R A ) and j < h 3j (R B ), j =, n (5) If the resource differences were very large, one player could overwhel the other on all sub-battlefields. To rule this out we assue that the resource difference between the players is not very large relative to the sub-battlefield advantages: j > h 1j (R A - R B ) and j > h 3j (R A - R B ), j =, n (6) In essence, allocating just the difference in resources available to the players to a battlefield is not enough to sway the outcoe of any non-neutral sub-battlefield. Note that (5) and (6) with respect to j iply that R A < 2R B. Next, if there is an exact balancing of the iportance of certain sub-battlefields, then knife-edge equilibria ay result. For exaple, if 2 = n 2, ultiple equilibria ay exist, because the gains and losses fro sall changes in allocations across the battlefields that only change the outcoes on the neutral sub-battlefields cancel out. 17 Note that other sub-battlefields could exist with very high advantages, for exaple, with kj > h kj (R B ) but those sub-battlefields will always be won by the favored player. This, they need not be considered by players when setting strategies. 16

18 Such equilibria are non-generic, arising only on easure zero sets in the paraeter space. The following condition rules out this type of equilibriu: 3 i1 ( t n s ) 0 when t i and s i each take any values fro the set (7) i i i i (0, 1, -1, ½, -½) but are not all zero. The expression in (7) is a general forula for the change in Vˆ due to changes in a player s strategy. Either a strategy change has no effect in a sub-battlefield, or it causes victory in the sub-battlefield to switch fro one player to the other, or it creates or breaks a tie in the sub-battlefield. One iplication of this assuption is that M N, so that the two battlefields cannot be exactly equal in iportance. Therefore, given the naing convention above, M < N ust hold. Another iplication is that n 2 2, so the iportance of the initially neutral sub-battlefields cannot be exactly equal. (III) Pure Strategy Equilibria In a Colonel Blotto gae with perfect targeting, a pure strategy equilibriu generally exists when there are only two battlefields, with both contestants expending all their resources in the larger battlefield. With iperfect targeting, even though the players have only two alternatives on which to spend resources, there are really ore than two battlefields and pure strategy equilibria typically do not exist. To analyze this, we begin with Lea 1, which rules out interior pure strategy equilibria. Lea 1: Given paraeter restrictions (5) (7), there do not exist any pure strategy equilibria with 0 < x < R A and 0 < y < R B. 17

19 To specify the conditions for pure strategy equilibria, consider the case when R A = R B R, and define the cutoff values y and y by h 3 (y) = and h 1n (R y) = h 1n (R) n. When x = 0, A wins sub-battlefield (3, ) if y < y and B wins if y > y, while A wins sub-battlefield (1, n) if y > y and B wins if y < y. Siilarly, let x and x be defined by h 1 ( x ) = and h 3n (R x) = h 3n (R) n. When y = 0, A wins sub-battlefield (1, ) if x > x and B wins if x < x, while A wins sub-battlefield (3, n) if x < x and B wins if x > x. Given assuption (5), y, y, x, and x all lie between 0 and R. Theore 1: Under assuptions (1) (7) and the convention that the battlefields are naed so that M < N, the only possible pure strategy equilibriu is x = y = 0. This equilibriu exists if and only if R A = R B, 2 < n 2, and both of the following conditions hold: (A) Either y < y and ½n 2 > ½ or y y and n 1 + ½n 2 > ½ (B) Either x < x and ½n 2 > 1 + ½ 2 or x x and ½n 2 + n 3 > 1 + ½ 2 Fro Theore 1, when each player has sufficient resources which differ fro those of the other player but not by too large an aount, then no pure strategy equilibriu exists. It never exists when R A R B. When R A = R B, it exists only under further restrictions on the paraeters of the odel. To gain soe intuition behind this result, recognize that given the budget constraint, each player has a one-diensional strategy space: how uch is allocated to battlefield. Fixing the opponent s allocation, 18

20 a player s payoff is nononotonic over this strategy space. As a player allocates ore resources toward, sub-battlefields in will switch allegiance to that player, while those in n switch away at different allocation levels. Because of this non-onotonicity, at least one player will have an iproving deviation over any pair of pure strategy allocations, one for each player. The following result helps in understanding the conditions deterining the relations between y and y and x and x, at least in one special case. Assue that effectiveness is the sae in the sub-battlefields of each battlefield, with h i (z) = h (z) and h in (z) = h n (z), all i and z. Further, assue that battlefield is less iportant than n because it is saller, and hence that allocations are ore effective there with h (z) h n (z), all z. Lea 2: If < n, or if = n and h j < 0, then y < y. If < n, or if = n and h < 0, then x < x. j Therefore, if the ore iportant battlefield n also has larger advantages ( n > and n > ), then fro Lea 2 and conditions (A) and (B) of Theore 1, ½n 2 > ½ and ½n 2 > 1 + ½ 2 ust hold at a pure strategy equilibriu. Together these iply that n 2 > M. Not only is battlefield n ore iportant overall, its neutral sub-battlefield ust be ore iportant than all of battlefield. In this case, both players put all their resources in the ore iportant battlefield and devote no resources to the less iportant one. If the advantage relations do not hold, the conditions on n 2 relative to the iportance of sub-battlefields in are still sufficient but are not necessary for the pure 19

21 strategy equilibriu to exist. It is worth noting that in soe settings, it is plausible to have larger advantages in the ore iportant battlefield. For exaple, consider the advertising exaple, where the ore iportant battlefield is siply the larger arket. The advantages can be interpreted as how uch resources ust be expended to shift an individual s decision. In larger counities, the price of an advertiseent ay be larger, so that ore ust be spent to have an individual see the sae nuber of ads. Pure strategy equilibria ay exist if one player s resources are significantly larger than the other with assuption (6) violated. For exaple, with enough resources, player A could set x = x n = ½R A and, no atter what B did, win all sub-battlefields in battlefield provided + h 1 (½R A ) h 1 (R B ) > 0, and all sub-battlefields in battlefield n if n + h 1n (½R A ) h 1n (R B ) > 0. Although the foral results in Theore 1 apply to a situation with only two battlefields, the intuition that pure strategy equilibria are unlikely to exist carries over to ore than two battlefields. For a pure strategy equilibriu to exist in a setting of K battlefields, the equilibriu strategies restricted to any pair ust be an equilibriu in that pair holding fixed allocations to all the other battlefields. Let R ij A and R ij B be the su of resources allocated to battlefields i and j by the two players. If the players have total resources which do not differ uch, then, on at least one pair of battlefields i and j, the resources R ij A and R ij B will satisfy assuptions (5) and (6) with an equilibriu then unlikely on the pair. (IV) Mixed Strategy Equilibria 20

22 In ost circustances, no pure strategy equilibriu exists and the players use ixed strategies. Finding the equilibriu ixed strategies in general is difficult. In Theore 2 below, we are able to find a necessary condition on the equilibriu ixed strategies under the assuption that players have the sae resources. To specify this, let E(x ) and F(y ) denote the cuulative distribution functions for the ixed strategies of A and B, respectively. As cdfs, E(x ) and F(y ) are nondecreasing and right-hand continuous and can be divided into continuous parts E*(x ) and F*(y ) and countable sets of values at which positive probability is placed. Let e a (x ) and f a (y ) denote the probability placed on the specific values x and y if they are ass points. Fro righthand continuity, E(x ) = li 0 E(x ) + e a (x ) and F(y ) = li 0 F(y ) + f a (y ). Consider the case in which the players have the sae resources R R A = R B. Consider sub-battlefield (3, ), which favors player A. A wins the sub-battlefield if + h 3 (x ) h 3 (y ) > 0. There exists an x such that + h 3 (x ) h 3 (R) > 0 iff x > x. Since x < R ust hold, there exists an ( ) = R - x with 0 < ( ). Directly, ( ) is defined by h 3 (R - ( )) h 3 (R). ( ) defines a lower bound on x such that A wins sub-battlefield (3, ) no atter how uch B spends in battlefield. That is, for any y and all x > R - ( ), + h 3 (x ) h 3 (y ) > 0. Siilarly, ( n ), defined by h 3n (R - ( n )) h 3n (R) n, gives a lower bound on x n such that for all x n > ( n ) and any y n, n + h 3n (x n ) h 3n (y n ) > 0. For sub-battlefields 1 of both battlefields, which lean to B, we can define ( ) by h 1 (( )) and ( n ) by h 1n (( n )) n. In this case, for all x < ( ) and any y, - + h 1 ((x ) h 1 (y ) < 0, so B definitely wins (1, ). Siilarly, for all x n < ( n ) and any y n, - n + h 1n (x n ) h 1n (y n ) < 0, so B definitely wins (1, 21

23 n). Since h 1i (R) > i by assuption (5), ( i ) < R ust hold. Let in[( ), ( n ), ( ), ( n )]. Fro the construction of ( i ) and ( i ), it iediately follows that < R. In addition, we assue R < 2 (8) Clearly, it follows fro (8) that R < ½ R <. If either player chooses an allocation in the interval (R, ), then, no atter what its opponent does, that player cannot win either of the sub-battlefields favorable to the opponent, but also cannot lose either subbattlefield in which that player is favored. Only the neutral sub-battlefields are in play. On the other hand, if an allocation is chosen outside the interval, there are potential gains and losses in the other sub-battlefields. For exaple, if x is below R thiseans that x n is above In this case, if y is large, A could lose sub-battlefield (3, ), while B could lose sub-battlefield (1, n). Given a value of R, this assuption in effect puts lower bounds on the agnitudes of the i and i. These advantages cannot be too large relative to R fro assuption (5). Fro (8), they also cannot be too close to 0. Note that (8) in at least soe circustances strengthens (6). If the h ij are linear, then ( i ) = c ij i and ( i ) = c ij i, for soe positive constants c ij so in(c 1, c 1n n, c 3, c 3n n ). Hence, for R A = R B, (6) only iposes > 0 while (8) iposes the ore restrictive > ½ R. Theore 2: Assue R A = R B and paraeter restrictions (5) (8). Either E() e a () = E(R ) or F() f a () = F(R ), so that at least one player never expends resources in the open interval (R ). 22

24 Although ultiple ixed strategy equilibria ay exist and it is difficult analytically to find the exact equilibria, Theore 2 gives an iportant characterization of any ixed strategy equilibriu, at least in circustances when the resources available to the two players are the sae. In no equilibria do both players have any probability of dividing their resources nearly equally between the two battlefields. This does not iply that even one of the players will, with positive probability, have an approxiately equal division of expenditure between battlefields. In fact, if the gae is syetric, then neither player will ever have an approxiately equal division. Corollary 1: If R A = R B = R, i = i, n 1 = n 3 and 1 = 3, then in any ixed strategy equilibriu E() e a () = E(R ) and F() f a () = F(R ). The significance of these results depends upon the size of the interval (R ). For near its lower bound of ½ R, the interval around ½ R is sall. Only allocations with alost exactly equal spending in the two battlefields are ruled out. For near its upper bound R, this interval is alost the entire set of allocations. Anything except for alost coplete asyetry in expenditures is ruled out. The size of this interval thus depends upon the agnitude of the initial advantages relative to the resources available to players. The ore resources players have relative to these advantages, the ore asyetric ust be the allocations of a player. Throughout the analysis, we have assued that the resources are exogenous and uncorrelated to the advantages i and i. However, there ay be reasons for the to be positively or negatively correlated in soe of the exaples of the odel. Consider the 23

25 case of political capaigns. When the advantages are greater voters are less persuadable, so oney will be less effective and donors ay prefer to donate to other candidates in other races, yielding a negative correlation. On the other hand, in larger battlefields, ore oney is raised but ads ay be ore expensive, which is siilar in effect to having larger advantages, creating a positive correlation. 18 In general, assue that a br where the sign of b deterines whether and R are positively or negatively correlated. Then the interval becoes ((1 b)r a, br + a) with the size of the interval 2a (2b 1)R. The size of this interval declines in R for b < ½ and increases for b > ½. Only if b is near ½ and a is near 0 will the interval be very sall for all values of R. Otherwise, for at least soe R, it should be non-negligible in size. Both Theore 2 and Corollary 1 are knife-edge results, holding for situations with exactly equal resources for the players or exactly syetric gaes. However, the essence of the results are likely to carry over to nearby situations with the resources of the players siilar but not exactly equal, or to alost syetric gaes. If the equilibriu correspondence of the gae is upper hei continuous, since every equilibriu at R A = R B has one player placing no weight near equal division, then for a neighborhood of resource pairs around R A = R B, at least one player would put little probability weight in the interval (R A ) or (R B ). Siilarly, for alost syetric gaes, both players would put little weight in those intervals. (V) Epirical Evidence in Political Capaigns 18 See Stratann (2005) for a survey of the recent capaign contributions and capaign spending literatures, where he discusses both of these effects. Stratann notes that when voter preferences are strong toward one candidate or the other, contributors are less likely to donate since capaign spending would have little effect in those arkets. He also discusses the fact that spending is likely to buy very different aounts of advertising in different arkets. 24

26 If the candidates behaviors are consistent with the Blotto approach, then at least one candidate will use very asyetric strategies across arkets, regardless of whether the equilibriu is in pure or ixed strategies. To test this, we exaine the data to see if candidates run substantially different nubers of advertiseents across edia arkets. Note that strategies are not copared across candidates, but across arkets for each candidate. Our advertising data include inforation only for the 100 largest broadcast arkets in the United States. We initially include in our analysis any part of a arket in a state with a political contest, even if only a tiny fraction of the arket is in the state in question. The iplications of this will be discussed below. We first consider states in which there are only two arkets or in which one arket is uch larger than all the rest of the arkets cobined. These are the situations ost consistent with the foral odel. The results above were derived for allocations between two arkets. When ore arkets exist but one is very large relative to the others, it can be shown that a pure strategy equilibriu will still exist with the candidates only running ads in the large arket. Table 1 lists each of the eleven states consistent with these criteria. Many edia arkets cross state borders, so that the arket ay be divided between two, three, or even ore states. The table includes all edia arkets, or parts of edia arkets, in the state, and lists the population of the arket that is within that state in question. For exaple, the Washington, D.C. edia arket falls not only in the District of Colubia proper, but also in the states of Maryland, Virginia and West Virginia. The population listed in the table for Washington, D.C. includes only those individuals in the D.C. arket who live in the state of Maryland. The fourth colun in the table, fraction of 25

27 state, shows the fraction of the state s broadcast arket population 19 that falls in the arket in question. For each state, these fractions should su to 1, with soe allowance for rounding. The final colun in the table, fraction of arket, shows the fraction of the arket that falls in the state listed. For exaple, all of the Baltiore edia arket is within the state of Maryland, but only 45% of the D.C. broadcast arket is in Maryland. That states ay have very sall parts of large edia arkets along with other arkets has interesting iplications for our analysis. If ad prices are proportional to arket population, and if two edia arkets have the sae fraction of their populations in a particular state, then the cost to a candidate of reaching a single voter is the sae in both arkets even if the arkets differ in size. However, if a state has a large fraction of one arket and a sall fraction of a second arket, then the relative costs of reaching a single voter is proportional to these fractions. First, consider the four states in our saple with only two arkets or arket portions: Maryland, New Jersey, New Mexico and Oregon. The arkets in Maryland are of approxiately equal size. Recall, however, that the Baltiore arket is entirely in Maryland whereas less than half of the Washington, D.C. arket is in Maryland. Thus, it is ore than twice as expensive to reach a Maryland voter by buying ads in the D.C. arket as in the Baltiore arket. The other three states have uch larger differences in arket size. In New Jersey, the New York arket is about three ties as large as the Philadelphia arket but the fractions of each arket in that state are about the sae, so the cost of reaching a single New Jersey voter fro each arket are very siilar. In New Mexico, the 19 We define the state s broadcast arket population as the population living within one of the 100 largest broadcast arkets. State residents living outside one of these broadcast arkets are not included in our totals. 26

28 Albuquerque arket is about nine ties the size of the El Paso arket. Only a fifth of the El Paso arket, but nearly all of the Albuquerque arket, is in New Mexico. This iplies that the cost of reaching a voter by an ad there is perhaps four and a half ties the cost in Albuquerque. Finally, in Oregon, the Portland arket has 228 ties the population of the Spokane arket. This very large difference is due to the fact that such a sall fraction (less than 1%) of the Spokane arket falls in the state of Oregon, whereas the entire Portland arket is in Oregon. Thus, the cost of reaching an Oregon voter fro Spokane is 100 ties that of reaching a voter fro Portland. Given this price differential, it would be consistent with alost any optiizing odel --- not just Blotto --- for Oregon candidates to place no ads in Spokane, and this is in fact what we see in the data. For this reason, we think that it is appropriate to drop Spokane as a arket in this state and hence to treat Oregon as having only one edia arket. Second, consider the seven states with ultiple arkets but where one arket is uch larger than all the other arkets in the state cobined: Arizona, Colorado, Georgia, Illinois, Nebraska, New Hapshire, and Veront. Several of the arkets in these states will have very high relative advertising prices --- ore than 20 ties that in another arket --- and will thus be dropped fro the analysis. These include the portion of the Albuquerque arket in Arizona (where the relative price would be 25 ties that in Phoenix or Tucson) and Colorado (where the relative price would be at least 24 ties that in Colorado Springs or Denver). By the sae logic, Greenville and Jacksonville are dropped as Georgia arkets, Evansville is dropped as an Illinois arket, and Nebraska 27