Proportional resource allocation in dynamic n-player Blotto games
aa r X i v : . [ ec on . T H ] O c t Proportional resource allocation in dynamic n -playerBlotto games ∗ Nejat Anbarcı † Kutay Cingiz ‡ Mehmet S. Ismail § October 13, 2020
Abstract
We introduce a novel and general model of dynamic n -player Blotto contests. Theplayers have asymmetric resources, the battlefields are heterogenous, and contest suc-cess functions are general as well. We obtain one possibility and one impossibilityresult. When players maximize the expected value of the battles, the strategy profilein which players allocate their resources proportional to the sizes of the battles at ev-ery history—whether their resources are fixed from the beginning or can be subject toshocks in time—is a subgame perfect equilibrium. However, when players maximizethe probability of winning, there is always a distribution of values over the battlessuch that proportional resource allocation cannot be supported as an equilibrium. JEL: C73, D72 ∗ We are grateful to Jean-Jacques Herings, Arkadi Predtetchinski, and J´anos Flesch without whosefeedbacks it would not be possible to complete this paper. We would like to thank Steven Brams, KangRong, Jaideep Roy, Christian Seel, C´edric Wasser, and audiences at Maastricht University, HEC Montreal,and Games and Contests workshop at Wageningen, King’s College London, and Durham University fortheir valuable comments. † Durham University Business School, Durham University, Durham DH1 3LB, UK. [email protected] ‡ Agricultural Economics and Rural Policy Group, Wageningen University, Hollandsweg 1, 6706KN Wa-geningen, The Netherlands. [email protected] § Department of Political Economy, King’s College London, London, WC2R 2LS, [email protected]
Introduction
Social, economic, and political interactions that resemble contests are ubiquitous. Ex-amples include rent-seeking, political campaigns, sports competitions, litigation, lobbying,wars and so on. These wide-ranging applications led economists theoretically model con-tests to understand the strategic incentives behind them. Traditionally, the first contestmodel of “Colonel Blotto” game was introduced by Borel (1921). A Colonel Blotto gameis a two-person static game in which each player allocates a limited resource over a numberof battlefields, each of which is of the same “size.” Fast forward, the literature on contestsis now enormous, and a Blotto contest denotes any contest in which two or more playersallocate a limited resource over some “battlefield.”In this paper, we introduce, to the best of our knowledge, the first model of dynamicmulti-battle n -player Blotto games in a very general framework as follows. The playerseach have possibly asymmetric resources, the battlefields are heterogenous, and the contestsuccess functions are quite general as well. In our benchmark model, each player starts thedynamic contest with a limited budget and distributes this budget over a finite numberof battlefields, which could be of varying sizes. For example, one battlefield may be thedouble or triple the area of another battle. Since the battles take place in a sequentialorder, players can condition their strategies on the outcomes of previous battles. At time t , players choose simultaneously their allocation on battle t to win x t >
0, which could beinterpreted as the area of the battlefield. The winner in a battlefield is determined by acontest success function satisfying the axioms (A1-A6) of Skaperdas (1996). The winnerand the resulting resource allocations are revealed to every player before the next battletakes place. The overall winner of this dynamic game is the player who wins the most area.We also introduce an extension of our model to the case in which the initial resources ofthe players might be subject to an exogeneous shock and can change throughout the game.The proportional allocation of resources is generally considered to be a benchmarkin resource distribution games, and especially in Blotto contests it is arguably one ofthe prominent strategies. In experimental static Blotto games, proportional distributionis usually considered to be a level-0 behavior. Studying a static, simultaneous-move For an experimental Blotto game see Arad and Rubinstein (2012); for level- k reasoining, see Stahl Brams and Davis (1974, p. 113) showedthat populous states receive disproportionately more investments with regards to theirpopulation. More specifically, the winner-take-all feature of the Electoral College—i.e.,that the popular-vote winner in each battle wins all the electoral votes of that battle—induces candidates to allocate campaign resources roughly in proportion to the 3 / (1993). As noted by Brams and Davis (1974), the population of a state need not exactly reflect the proportionof the voting-age population who are registered and actually vote in a presidential election. Relevant Literature
Our paper contributes to the literature on dynamic contests and campaign resource allo-cation in sequential elections. This brief section summarizes related work apart from theones mentioned in the Introduction.In a recent closely related paper, Klumpp, Konrad, and Solomon (2019) consider dy-namic Blotto games where two players fight in odd number of battle fields, which areidentical. The player who wins the majority of battles wins the game. Accordingly, theyshow that under general contest success functions players allocate their resources evenlyacross battles in all subgame perfect equilibria, one of which is in pure strategies. Theirresult generalizes our two-player example with three symmetric battlefields presented insection 4.Note that their very interesting finding does not contradict our preceding impossibilityresult because they study two-player symmetric dynamic contests where battlefields are ofthe same size.Sela and Erez (2013) studied a two-player dynamic Tullock contest, in which each playermaximizes the sum of the expected payoffs (similar to expected value maximization in oursetting) for all districts. Suppose that the value of the battles is equal across the stagesand that for each resource unit that a player allocates, he loses 0 ≤ α ≤ In a similarsetting, Deck, Sarangi, and Wiser (2017) have recently studied symmetric static contestswith two players who do not have budget constraints. They identified the Nash equilibrium Colonel Blotto games were first introduced by Borel (1921). Among others, recent contributions toBlotto games include Roberson (2006), Kvasov (2007), and Rinott, Scarsini, and Yu (2012). There is also ahuge literature on non-Blotto contests initiated by Tullock (1967; 1974), and see also, e.g., Krueger (1974).The early literature on non-Blotto contests are motivated by rent-seeking. For a discussion of dynamics in contests, see Konrad (2009). For experimental results on Blotto games see, e.g., Deck and Sheremeta (2012) and Duffy and Matros(2017) and the references therein.
3f the symmetric game (Electoral College).In another static presidential campaign model, Lake (1979) argued that one would needto assume that the candidates maximize only their probability of winning the election, i.e.,one would simply try to receive a majority of electoral votes, instead of complying withBrams and Davis’ (1973, 1974) assumption that they maximize their expected electoralvote. Nevertheless, Lake’s (1979) main result echoes Brams and Davis’ (1974) impossibilityof population proportionality result in that in Lake’s model too it turns out that presidentialcandidates find it optimal to spend a disproportionately large amount of their funds in thelarger states.Harris and Vickers (1985) construed a patent race as a multi-battle contest, in whichtwo players alternate in expending resources in a sequence of single battles. These battlesor sub-contests serve as the components of the overall R&D contest. Just like in a singlestennis match, the player who is first to win a given number of battles wins the contest, byobtaining the patent. Additional work on dynamic resource allocation contests is as follows. Dziubinski,Goyal and Minarsch (2017) have recently studied multi-battle dynamic contests on net-works in which neighboring ‘kingdoms’ battle in a sequential order. Hinnosaar (2018)chracterizes equilibria of sequential contests in which efforts are exerted sequentially towin a (single-battle) contest. In a two-player and two-stage campaign resource allocationgame, Kovenock and Roberson (2009) characterized the unique subgame perfect equilib-rium. In a two-player best-of-three multi-battle dynamic contest, Konrad (2018) anaylzesresource carryover effects between the battles. Brams and Davis (1982) examined a modelof resource allocation in the U.S. presidential primaries to study the effects of momen-tum transfer from one primary to another. Klumpp and Polborn (2006) also focused onmomentum issues; they considered a two-player model in which an early primary victoryincreases the likelihood of victory for one player and creates an asymmetry in campaignspending, which in turn magnifies the player’s advantage. This asymmetry of campaign In the PGA Tour, which brings professional male golfers together to play in a number of tournamentseach year (LPGA does so for female golfers), each tournament consists of multiple battles in that golfersattempt to minimize the total number of shots they take across 72 holes.
We consider dynamic Blotto contests where there are m heterogeneous battle fields, indexedby t = 1 , , ..., m , and n players, indexed by i = 1 , , ..., n . The battles take place in apredetermined sequential order. Players have possibly asymmetric budgets: Each player i has a budget W i ≥ t , the valueof the battle is denoted by x t >
0, which could be interpreted as the “area” of the battlefield t . Each time period t , the battle at t takes place, and each player i simultaneously choosesa pure action (allocation) denoted by w ti which is smaller than or equal to the budget, W i ,minus the already spent allocation by player i until battle t . Given the chosen actions inbattle t , w t := ( w t , . . . , w tn ), the probability of player i winning battle t is defined by acontest success function, which has the following form. p ti ( w t ) = f ( w ti ) P j f ( w tj ) if P j w tj > n if P j w tj = 0 , (3.1)where f ( . ) satisfies Skaperdas’s (1996) axioms (A1-A6), which characterizes a wide rangeof contest success functions used in the literature. More specfically, it is of the followingform: f ( w tj ) = β ( w tj ) α for some α > β > t , x t < P t ′ = t x t ′ , that is, there is no“dictatorial” battle. Let x ti be the value player i wins at battle t , which is x t with probability p ti ( w t ) or 0 with probability 1 − p ti ( w t ).The set of histories of length t is denoted by H t . A history of length t ≥ h t := ((( w , x ) , . . . , ( w n , x n )) , . . . , (( w t , x t ) , . . . , ( w tn , x tn ))) (3.2)satisfying the following conditions(i) For each 1 ≤ i ≤ n and for each 1 ≤ t ′ ≤ t , w t ′ i ∈ [0 , W i − P j
Now suppose that players maximize the expectedvalue they get from the battles (i.e., the sum of battlefield areas), so the terminal historiesare exactly the histories with length m . The set of terminal histories is denoted by H m ,which is equal to H m . 6or any ¯ h m ∈ H m , player i receives a payoff equal to¯ u i (¯ h m ) = X t ≤ m x ti , (3.3)where ¯ h m := ((( w , x ) , . . . , ( w n , x n )) , . . . , (( w m , x m ) , . . . , ( w mn , x mn ))).Analogous to the maximizing-probability-of-winning case, the set of terminal historiesinduced by a strategy profile σ conditional on reaching history h is denoted by ρ ( σ | h ),which is a subset of H m . The payoff for player i ≤ n induced by a pure strategy profile σ ∈ Σ at any h t ∈ H t is v i ( σ | h t ) = X ¯ h m ∈ ρ ( σ | h t ) q ( σ, ¯ h m | h t )¯ u i (¯ h m ) . (3.4) Maximizing probability of winning:
A player wins the multi-battle dynamic contestif he or she wins the most battlefield areas. In the context of sequential elections, thiswinning rule is equivalent to the plurality rule in which the candidate who receives theplurality of votes wins the election. In this part, we assume that players maximize theprobability of winning.An element ¯ h t ∈ H is called terminal if the contest ends with battle at battle t —ifeither t = m or there exists a player i such that X j ≤ t x ji > max { X j ≤ t x ji ′ | i ′ = i } + x t +1 + . . . + x m . (3.5)If at some history a player is guaranteed to lose, then the player’s remaining budgetafter that history is 0. Thus players who guaranteed to lose stay in the game and spend0 at the remaining battles. Furthermore, if at some history a player is already guaranteedto win, then the contest ends at this history.Let H = S t ≤ m H t be the set of all terminal histories. For any given ¯ h t ∈ H , let C (¯ h t ) bethe set of players that have won the highest number of battlefield areas up to and includingbattle t , which is defined by C (¯ h t ) = arg max i ≤ n P j ≤ t x ji . For ¯ h t ∈ H , player i receives apayoff equal to u i (¯ h t ) = | C (¯ h t ) | if i ∈ C (¯ h t )0 otherwise . (3.6)7or every t ≤ m , we define ρ : Σ × H t → P ( H ) where P ( H ) is the power set of H such that ρ ( σ | h t ) denotes the set of terminal histories that are reached with positive probability withrespect to σ conditional on reaching history h t ∈ H t . The probability of a terminal history¯ h being reached with respect to σ conditional on reaching h t is denoted as q ( σ, ¯ h | h t ). Thepayoff for player i ≤ n induced by a pure strategy profile σ ∈ Σ at any history h t ∈ H t isdefined as v i ( σ | h t ) = X ¯ h ∈ ρ ( σ | h t ) q ( σ, ¯ h | h t ) u i (¯ h ) , (3.7)We denote v i ( σ | ø) by v i ( σ ). Equation (3 .
7) and q ( σ, ¯ h | h t ) define the payoff of each player i as follows. For any given terminal history ¯ h in ρ ( σ | h t ), we multiply the probability ofreaching the given history ¯ h with the utility player i gets at terminal history ¯ h and thenwe sum over all terminal histories in ρ ( σ | h t ).Our solution concept is subgame perfect equilibrium in pure strategies. Subgame perfect equilibrium:
A pure strategy profile σ ∈ Σ is a subgame perfectequilibrium if for every battle t ≤ m , for every history h ∈ H t , for every player i ≤ n, andfor every strategy σ ′ i ∈ Σ i v i ( σ | h ) ≥ v i ( σ − i , σ ′ i | h ) . A strategy profile σ ∈ Σ is a subgame perfect equilibrium if and only if for every h ∈ H, σ induces an equilibrium in the subgame starting with history h .We will also need the following definition: Proportional strategy profile:
Here we define a very specific pure strategy profile σ .For any t , for any nonterminal history h t − ∈ H − H , and for any player i , let σ ti ( h t − ) = B i ( h t − ) x t x t + . . . + x m . (3.8)We call σ the proportional pure strategy profile. Note that under σ , no matter what theother players do, every player proportionally allocates his or her available budget over theremaining battles. A dynamic contest is said to admit proportionality if the proportionalstrategy profile is a subgame perfect equilibrium.8 Dynamic Blotto games where players maximize theprobability of winning
In this section, the players maximize their probability of winning the dynamic contest.Here we analyze three examples of dynamic contests, which will help establishing our mainresult in the section while also shedding some light on the specifics of the dynamics of ourcontest: I) 3-identical-battle dynamic contest. II) 4-battle dynamic contest where the firstbattle is larger (i.e., has a higher value) than the 3 other identical battles. III) 4-battledynamic contest where the last battle is larger than the first 3 identical battles. We showthat the dynamic contest in Example I satisfies proportionality. Furthermore, we providenontrivial dynamic contests in Examples II and III that do not admit proportionality.Finally, we show that for any n -player dynamic contest with at least four battles andat least two players, there is always a distribution of values over the battles such thatproportionality does not satisfy. Suppose that there are two players A and B with equal budget W = 100 and three identicalbattles. To solve this dynamic contest, we use backward induction. If for given h , V ( h )is equal to ( A, A ) or (
B, B ), which means player A has already won or lost the first twobattles, then the contest comes to an end and player A wins or loses, respectively. If forgiven h , V ( h ) is equal to ( A, B ) or (
B, A ) then the dynamic contest continues to thelast battle and in the subgame after h , the unique best response is to allocate all of theremaining budget to the last battle. Therefore, in any subgame after any nonterminalhistory h the strategy profile ( σ A , σ B ), which is to spend the remaining budget to the lastbattle, is the unique Nash equilibrium.Now, suppose that for h , V ( h ) = ( A ). Then, player A ’s best response to B is tomaximize his payoff v A ( σ | h ) = v A ((( σ A , σ B ) , ( σ A , σ B ) , ( σ A , σ B )) | h ) (4.1)9ith respect to σ A . Given a fixed σ B , player A solves the following maximization problemmax σ A v A ( σ | h ) = max σ A ( h ) ( p A ( σ A ( h ) , σ B ( h ))+ (1 − p A ( σ A ( h ) , σ B ( h ))) × p A ( B A ( h ) − σ A ( h ) , B B ( h ) − σ B ( h ))) (4.2)= max σ A ( h ) ( σ A ( h ) σ A ( h ) + σ B ( h ) + σ B ( h ) σ A ( h ) + σ A ( h ) × − w A − σ A ( h )200 − w A − σ A ( h ) − w B − σ B ( h ) ) (4.3)= max w A ( w A w A + w B + w B w A + w B × − w A − w A − w A − w A − w B − w B ) . (4.4)Equality (4 .
2) holds, since player A already won the first battle, in order to win the contestplayer A either needs to win the second battle—given ¯ h with V (¯ h ) = ( A, A )—, or if heloses the second then to win the third battle—given h with V ( h ) = ( A, B ). Moreover, ifthe game continues to the last battle, then spending the remaining budget is the unique bestreponse in the subgame after given history. Equality (4 .
3) follows from the contest successfunction. Equality (4 .
4) follows from the definition of a strategy which maps histories toactions in the following battle.As player A wants to maximize v A ( σ | h ), player B wants to minimize it. We derivebest response functions from the first-order condition of v A ( σ | h ) with respect to σ A ( h )and σ B ( h ), respectively. The intersection of best responses provides us the following twoconditions w A = 12 (100 − w A ) , (4.5) w B = 12 (100 − w B ) . (4.6)So far, we have showed that the pair ( σ A , σ B )—in which players each allocate the remainingbudget to the last battle—, and the pair (( σ A , σ B ),( σ A , σ B ))—in which strategies satisfy(4 .
5) and (4 . Detailed calculations are available in the Appendix. = ø. Then, player A best responds to B by maximizing his payoff v A ( σ | ø) = v A ( σ ) withrespect to σ A (ø) max σ A (ø) v A ( σ ) = max σ A (ø) ( p A ( w )( p A ( w ) + (1 − p A ( w )) p A ( w ))+ (1 − p A ( w )) p A ( w ) p A ( w )) . (4.7)where w = ( σ A (ø) , σ B (ø)), and w , w are elements of [0 , − w A ] × [0 , − w B ] and[0 , − w A − w A ] × [0 , − w B − w B ], respectively. But since we have already concludedthat whoever wins the first battle, both players should invest equally all their remainingbudget to the last two battles by equations (4 .
5) and (4 . w A = w A =(1 / − w A ) and w B = w B = (1 / − w B ). Thus, we can rewrite equation (4 .
7) asmax σ A (ø)= w A ( w A − w A ) + 3( w A )( w B − − w B )( w A + w B − ( w A + w B ) . (4.8)Equations (4 .
7) and (4 .
8) follows from the analogous reasoning for equations (4 . .
3) and(4 . v A ( σ | h ), the first order condition of v A ( σ ) with respectto σ A (ø) and σ B (ø) yields the best response functions of players A and B . The intersectionof the best responses leads to the condition w A = w B = 100 /
3. From equations (4 . . w A = w B =100/3, we deduce that w A = w A = w A = w B = w B = w B = 100 /
3, whichconcludes that every subgame perfect equilibrium strategy profile of Example I satisfiesproportionality. This example is similar to the benchmark example of Fu, Lu, and Pan (2015, p. 9); though, oursettings differ when the battles are not identical. .2 Examples II and III: Dynamic contests with 4 battles whereone is larger than the others To illustrate our model with a nontrivial example, suppose that there are two players A and B with equal budget W = 100 and 4 battles, one of which is a 2-value battle and theother battles each have 1 value as presented below.Battle Example II : Let the 2-value battle be the first battle. If for given h , V ( h ) is equalto ( A, A ) or (
B, B ), which means player A has already won or lost the first two battles,then the contest comes to an end and player A wins or loses, respectively. If for given h , V ( h ) is equal to ( A, B, B ) or (
B, A, A ), then the dynamic contest continues to the lastbattle. And, for the subgame after the given history the unique best response for eachplayer ( σ A and σ B ) is to allocate all the remaining budget to the last battle.For given h , V ( h ) = ( A, B ), player A best responds to B by maximizing his payoff v A ( σ | h ) with respect to σ A max σ A v A ( σ | h ) = max σ A ( h ) v A ( σ | h ) = max w A ( p A + (1 − p A ) p A ) , which is analogous to the dynamic contest with 3 identical battles. Hence, each playerallocating equally her remaining budget to battles 3 and 4 is the unique best response,( σ A , σ A ) and ( σ B , σ B ), for subgames starting with history h . Now suppose that for given h , if V ( h ) = ( A ), then player A best responds to B by maximizing his payoffmax σ A ( h ) v A ( σ | h ) = max w A ( p A + (1 − p A ) p A + (1 − p A )(1 − p A ) p A ) . (4.9) Calculations are available upon request. From this point on, we take p ti as p ti ( w t ).
12e derive Equation (4 .
9) by an analogous method from Example I. To obtain the bestresponse functions, we take the first order conditions of the payoff functions with respect to σ A ( h ) and σ B ( h ), respectively. The intersection of the best responses yields the followingtwo conditions w A = 13 (100 − w A ) , (4.10) w B = 13 (100 − w B ) . (4.11)Given that each player wins one battle from the first two battles, we already concludedthat players allocate their remaining budget to battles 3 and 4 equally. Thus, by equa-tions (4 .
10) and (4 . h = ø. Then, player A best responds to B by maximizing hispayoff v A ( σ | ø) = v A ( σ ) with respect to σ A (ø) which is max σ A (ø) v A ( σ ), i.e.,max σ A (ø) v A ( σ ) = max w A ( p A ( p A + (1 − p A ) p A + (1 − p A )(1 − p A ) p A ) + (1 − p A ) p A p A p A )= ( w A − w A ) ( w B − w A ) + 10000 w B )( w A + w B − ( w A + w B )+ ( w A − w A (3( w B ) − w B + 70000))( w A + w B − ( w A + w B ) (4.12)Equation (4 .
12) is derived from a similar method applied to equation (4 . .
12) with respect to w A and w B yields the best response functions ofplayers A and B . We deduce that w A = w B = 50 is at the intersection of the best responses.Hence, equations (4 .
10) and (4 .
11) imply that w A = w A = w A = w B = w B = w B = 50 / σ such that onan equilibrium path of σ , players spend half of their budget to the 2-value battle and thenequally split the remaining budget among 1-value battles. Thus, Example II does notadmit proportionality. Example III : Let the 2-value battle be the last battle as presented below.13attle h , V ( h ) is equal to ( A, B ) or (
B, A ), then players allocating all their remainingbudget to the 2-value battle is the unique best response for the subgame starting fromhistory h . If for given h , V ( h ) is equal to ( A, A ), then in the following subgame of therelated history, player A needs only one of the remaining battles and player B needs bothbattles to win the contest which is the same case as in Example I after the first battle.Thus, the unique best responses are players distributing equally their remaining budgeton the last two battles. Therefore, in any subgame after any history h ∈ H , there is noequilibrium that is proportional. Hence, Example III does not admit proportionality.The following proposition provides a class of dynamic contests that does not admitproportionality whenever players maximize probability of winning. Proposition 1
For any contest success function `a la Skaperdas (A1–A6), any m ≥ number of battles and any n ≥ players, there exists a dynamic contest, where playersmaximize their probability of winning, for which proportionality fails. Proof:
Consider a dynamic contest such that x = . . . = x m − = 1 and x m = 3. Considerthe history h m − where player 1 wins battle 1, player 2 wins battle 2, player 1 wins battle3 and so on up to and including battle m −
2. Thus, for history h m − , the maximum dif-ference between the total values of player 1 and player 2 is 1. In the subgame after history h m − , player 1 and player 2 spending all their budget to the last battle is the unique Nashequilibrium. Since we reach history h m − with positive probability under the proportionalstrategy profile, the dynamic contest does not admit proportionality. (cid:3) This simple impossibility result shows that Klumpp, Konrad, and Solomon’s (2019)result on two-player symmetric dynamic Blotto games with same-size battlefields does notextend to more general dynamic contests with heterogenous battlefields. To gain somefurther insight, let’s consider the following example.Battle
In this section, we assume that players maximize the expected value that they receivefrom the battlefield in the n -player sequential multi-battle contest. We first define thewell-known Tullock contest success function in this context. p ti ( w t ) = w ti P j w tj if P j w tj > n if P j w tj = 0 . (5.1)The following theorem provides our second main result. Note that in the followingsection we generalize our model and extend this result to more general contest successfunctions. Theorem 2
For any dynamic contest with Tullock contest success function where playersmaximize winning the expected value of the battles, proportionality is satisfied.
Proof:
We show that the proportional strategy profile σ = ( σ i ) i ≤ n is robust to one-shotdeviations, which implies that σ is a subgame perfect equilibirum. That is, any player i
15t any nonterminal history h t can not improve his payoff by changing σ ti , given that allother players, j = i , follow the proportional strategy. If player i switches to a strategy¯ σ i = (¯ σ t +1 i , σ t +2 i , . . . , σ mi ) after history h t such that ¯ σ t +1 i ( h t ) = σ t +1 i ( h t ), then the expectedvalue player i wins after battle t given the history h t is denoted as v i,t +1 (¯ σ i , σ − i | h t ), whichsatisfies v i,t +1 (¯ σ i , σ − i | h t ) = x t +1 ¯ σ t +1 i ( h t )¯ σ t +1 i ( h t ) + P j = i σ t +1 j ( h t ) + v i,t +2 ( σ | h t +1 dev ) , (5.2)where h t +1 dev is a successor of h t with the property that at battle t + 1 player i spent ¯ σ t +1 i ( h t ),and each player j = i spent proportionally. And the expected value of player i after history h t if she follows σ , v i,t +1 ( σ | h t ) = x t +1 σ t +1 i ( h t ) P ≤ j ≤ n σ t +1 j ( h t ) + v i,t +2 ( σ | h t +1 ) , (5.3)where h t +1 is a successor of h t with the property that at battle t + 1, each player spentproportionally. For simplicity we take x t +1 + . . . + x m x t +1 = k,B i ( h t ) = a, X j = i B j ( h t ) = b, ¯ σ t +1 i ( h t ) = σ t +1 i ( h t ) + ∆ = ak + ∆ . where ∆ is a real number. We can rewrite player i ’s probability of winning battle t + 1 ifhe plays ¯ σ t +1 i ( h t ) as ¯ σ t +1 i ( h t )¯ σ t +1 i ( h t ) + P j = i σ t +1 j ( h t ) = ak + ∆ ak + ∆ + bk = a + ∆ ka + ∆ k + b , and player i ’s probability of winning battle t + 1 if he plays σ t +1 i ( h t ) as σ t +1 i ( h t ) P ≤ j ≤ n σ t +1 j ( h t ) = akak + bk = aa + b . Since σ is a proportional strategy profile, for any t , for any h t , and for successor of historieswhere h t +1 is a successor of h t , h t +2 is a successor of h t +1 , and so on up to and including h m is a successor of h m − , given that players follow proportional strategy profile, we have σ t +1 i ( h t ) P ≤ j ≤ n σ t +1 j ( h t ) = σ t +2 i ( h t +1 ) P ≤ j ≤ n σ t +2 j ( h t +1 ) = . . . = σ mi ( h m − ) P ≤ j ≤ n σ mj ( h m − ) = aa + b , i wins each battle after h t with equal probability if he/she follows σ i . That is, if players follow the proportional strategy profile, the proportions of theremaining budgets stay constant throughout the battles. The same property satisfies forthe strategy profile (¯ σ i , σ − i ) after history h t +1 dev . Hence for successor of histories where h t +2 dev is a successor of h t +1 dev , h t +3 dev is a successor of h t +2 dev , and so on up to and including h mdev is asuccessor of h m − dev , given that players follow proportional strategy profile after history h t +1 dev ,we have σ t +2 i ( h t +1 dev ) P ≤ j ≤ n σ t +2 j ( h t +1 dev ) = σ t +3 i ( h t +2 dev ) P ≤ j ≤ n σ t +3 j ( h t +2 dev ) = . . . = σ mi ( h m − dev ) P ≤ j ≤ n σ mj ( h m − dev ) . Now we can simply calculate player i ’s probability of winning any battle after history h t +1 dev ,if player i follows the strategy ¯ σ i σ t +2 i ( h t +1 dev ) P ≤ j ≤ n σ t +2 j ( h t +1 dev ) = a − ak − ∆ a − ak − ∆ + b − bk . Therefore we can rewrite equation (5 .
2) as v i,t +1 (¯ σ i , σ − i | h t ) = x t +1 a + ∆ ka + ∆ k + b + a − ak − ∆ a − ak − ∆ + b − bk ( x t +2 + . . . + x m ) , And we can rewrite equation (5 .
3) as v i,t +1 ( σ | h t ) = aa + b ( x t +1 + . . . + x m ) . We show that v i ( σ | h t ) − v i (¯ σ i , σ − i | h t ) ≥
0, in other words show that x t +1 ( aa + b − a + ∆ ka + ∆ k + b ) + ( x t +2 + . . . + x m )( aa + b − a − ak − ∆ a − ak − ∆ + b − bk ) ≥ . (5.4)Since k − x t +2 + . . . + x m ) / ( x t +1 ), we can rewrite inequality (5 .
4) as( aa + b − a + ∆ ka + ∆ k + b ) + ( k − aa + b − a − ak − ∆ a − ak − ∆ + b − bk ) ≥ . (5.5)We can simplify inequality (5 .
5) as b ∆ k ( a + b )( a ( k −
1) + b ( k − − ∆ k )( a + b + ∆ k ) ≥ . (5.6)Inequality (5 .
6) satisfies because we have the following conditions a ≥ ak + ∆ , ( k − ≥ ,ak + ∆ ≥ . Thus, for any ∆ we have v i ( σ | h t ) − v i (¯ σ i , σ − i | h t ) ≥ (cid:3) We next present extensions of this theorem to a more general setting with variablebudgets and with any contest success function satisfying Skaperdas’s (1996) axioms A1–A6.
In this section, we show that our results are robust to exogenous budget shocks during thedynamic contest. We call this extension dynamic contests with variable budgets.
Changes in the model : A history of length t ≥ h t := ((( w , x ) , . . . , ( w n , x n )) , . . . , (( w t , x t ) , . . . , ( w tn , x tn ))) (6.1)satisfying the following conditions(i) For each 1 ≤ i ≤ n and for each 1 ≤ t ′ ≤ t , w t ′ i ∈ [0 , W i + z t ′ i − P j For any t , for any nonterminal history h t − ∈ H − H ,and for any player i , let σ ti ( h t − ) = ˆ B i ( h t − ) x t x t + . . . + x m . (6.2)Note that for any subgame starting at history h t − and for any player i , the knowledge ofplayer i on the variable budget set is { z t , . . . , z tn } . Therefore the strategy σ is defined onˆ B i ( h t ) but not on future variable budgets, which is exogenously given each time before abattle takes place. We call σ the proportional (pure) strategy profile. Note that under σ ,no matter what the other players do, every player proportionally allocates her availablebudget over the remaining battles.A dynamic contest is said to satisfy proportionality if the proportional strategy profileis a subgame perfect equilibrium. Mutatis mutandis, the rest of the model remains thesame as the model presented in section 3.The following proposition extends Proposition 1 to the case with variable budgets. Itsproof echoes the one of Proposition 1, and hence it is not reproduced. Proposition 3 For any contest success function `a la Skaperdas (A1–A6), for any m ≥ number of battles and any n ≥ players, there exists a dynamic contest with variablebudgets, where players maximize their probability of winning, for which proportionalityfails. We now state our last main result, which extends Theorem 2 to the case with variablebudgets and general contest success functions. Theorem 4 For any dynamic contest with variable budgets where players maximize theexpected value of the battles, the dynamic contest satisfies proportionality. roof: First, we show that a strategy profile σ ∗ ∈ Σ that satisfies proportionality in thedynamic contest is a Nash equilibrium by proving that for any player i a strategy σ i ∈ Σ i is a best response to σ ∗− i whenever σ i = σ ∗ i with fixed budgets, and then we extend thisresult to contests with variable budgets. Let w be the spending sequence associated with( σ i , σ ∗− i ) and w − i = ( w t − i ) t ≤ m denote the spending sequence excluding player i , where forall t , w t − i = ( w t , . . . , w ti − , w ti +1 , . . . , w tn ). We show that σ i ∈ arg max σ ′ i v i ( σ ∗− i , σ ′ i | ø) , (6.3)that is, player i ’s best response to σ ∗− i associated with w − i is σ i associated with w i . Giventhat all players but i follow the spending sequence w − i , player i ’s expected value from a1-value battle t for any w t i , which we treat as a variable, is given by β ( w t i ) α ( β ( w t i ) α + β P j = i ( w t j ) α ) . (6.4)Differentiating (6.4) with respect to w t i gives α ( w t i ) α − P j = i ( w t j ) α (( w t i ) α + P j = i ( w t j ) α ) , (6.5)which is player i ’s marginal gain from the battle t . We next consider a k -value battle t k forsome k ∈ { , ..., m } . By our supposition each player except player i spends in proportionto the value of the battle, i.e., for each j = i , w t k j = kw t j . We next show that player i ’s bestresponse to proportional allocation is also to spend in proportion to the value at battle t k ,i.e., w t k i = kw t i . In this case, player i ’s expected value for any w t k i from k -value battle t k is kβ ( w t k i ) α ( β ( w t k i ) α + β P j = i w t k j ) = k ( w t k i ) α (( w t k i ) α + k α P j = i ( w t j ) α ) . (6.6)Differentiating (6.6) with respect to w t k i gives αk α +1 ( w t k i ) α − P j = i ( w t j ) α (( w t k i ) α + k α P j = i ( w t j ) α ) , (6.7) To be sure, one may condition his strategy on the winners of the previous battles and also on theprevious battle spendings. However, without loss of generality, we can confine attention to the spendingsequence, w , that is associated with the given strategy profile, because the utility received from the previousbattles does not affect the utility that can be received from the remaining ones as the utility function isadditive. This is, of course, not true under winning probability maximization because winning a battlealters the ways in which the dynamic contest can be won. i ’s marginal gain from the k -value battle t k . Next, we show that for w t k i = kw t i , Expression 6.7 equals Expression 6.5. First, Expression 6.7 equals αk α +1 ( w t k i ) α − P j = i ( w t j ) α (( w t k i ) α + k α P j = i ( w t j ) α ) = αk α +1 ( kw t i ) α − P j = i ( w t j ) α (( kw t i ) α + k α P j = i ( w t j ) α ) . (6.8)Cancelling out k ’s leads to α ( w t i ) α − P j = i ( w t j ) α (( w t i ) α + P j = i ( w t j ) α ) , (6.9)which is Expression 6.5. We showed that if player i allocates proportionally to k -valuebattle, then his marginal gain from that battle is equal to his marginal gain from 1-valuebattle provided that others allocate proportionally. Thus, there is no incentive for player i to deviate from proportional allocation when others allocate proportionally. Hence, theproportional strategy profile σ is a Nash equilibrium of the dynamic contest.Now we show that σ ∗ is a subgame perfect equilibrium. In other words, for every h ∈ H, σ ∗ induces an equilibrium in the subgame starting with history h . By definition,the subgame starting with history h is a game (i.e., dynamic contest) and ( σ ∗ | h ) is aproportional strategy profile. Thus, by an analogous argument used in the first part of theproof, ( σ ∗ | h ) is a Nash equilibrium in the subgame starting with history h . That is, weobtain for every i and every h ( σ ∗ i | h ) ∈ arg max σ ′ i v i ( σ ∗− i , σ ′ i | h ) , (6.10)Therefore, σ ∗ is a subgame perfect equilibrium, so the dynamic contest satisfies propor-tionality.So far, we have shown that in the beginning of the dynamic contest with fixed budgetsplayers allocating their budgets proportionally is a subgame perfect Nash equilibirum ofthe dynamic contest. It implies that in the dynamic contest with variable budgets, playersallocating their budgets proportionally is also a Nash equilibirum in the beginning of thegame, because the initial budgets and payoffs are the same in the begining. After thecompetition in the first battle, players’ budgets are updated, which defines a new dynamic Note that in the first part we showed that a proportional strategy profile is a Nash equilibrium in anydynamic contest. (cid:3) In this paper, we have introduced a novel model of dynamic n -player Blotto games. Thisis a very general framework as the budgets and battlefields are possibly asymmetric, andthere are no restrictions on the number of players or the number of battlefields (e.g., oddor even). In this framework, we studied the proportional allocation of resources and theequilibrium behavior. We showed that the strategy profile in which players allocate theirresources proportionally at every history is a subgame perfect equilibrium whenever play-ers maximize the expected value of the battles. By contrast, when players maximize theprobability of winning the dynamic contest, there is always a distribution of values overthe battles such that proportionality cannot be supported as an equilibrium. We haveshown that our results are robust to exogeneous shocks on budgets. Moreover, the resultsdo not depend on the specfic contest success function used in the competition as longas it satisfies Skaperdas’s (1996) axioms (A1–A6). While we have focused on proportionalequilibrium behavior in this paper, we believe that characterizing non-proportional equilib-rium behavior—especially when players maximize their probability of winning—in n -playerdynamic Blotto contests is a promising future research direction.As we have mentioned, Blotto games can be applied to many economic and politicalsituations as Borel (1921) himself envisioned. As an example, consider sequential elec-tions as an n -player dyanmic multi-battle contest where political candidates choose howthey distribute their limited resources over multiple “battlefields” or states as in the U.S.presidential primaries. In this context, our results imply that proportionality is immedi-ately rectified once one has candidates who maximize their electoral vote instead of simplymaximizing their probability of winning, despite the presence of the winner-take-all feature.22o achieve proportionality in at least the U.S. presidential primaries which have thewinner-take-all feature, a viable policy suggestion could be to provide additional incentivesfor players to induce them to win as many delegates as possible in the entire presidentialprimaries overall. For instance, the electoral system may provide them additional fundingin the ensuing presidential race where these incentives are positively linked to the numberof delegates won by the presidential player in the primaries. Such incentives can be veryeffective at the margin. As a matter of fact, even in the absence of any such additionalpecuniary incentives, for one reason or another, players seem to already have the behavioraltrait of maximizing their expected delegates themselves and do not want to stop pumpingcampaign funding to their remaining primaries even though they have already guaranteedwinning the majority of the delegates in the primaries. The main reason that the playersmay try win more delegates beyond what they would need to guarantee their presidentialcandidacy (i.e., the main reason that they still might keep investing in the remainingprimaries even though they know that it will not affect their chances of winning anyfurther delegates) could be that they care about entering the U.S. presidential race withan impressive momentum gained in the presidential primaries, as Hillary Clinton tried todo against the late surge of Bernie Sanders in the U.S. Democratic primaries in 2016 eventhough she had already accumulated more than sufficiently many delegates to win herparty’s presidential candidacy up to that point. Nevertheless, to ensure proportionality,the parties or the electoral system may consider boosting players’ tendency to maximizetheir expected delegates via some additional pecuniary incentives, which may help at themargin at least for the players who may simply try to maximize their winning probabilityin their U.S. presidential primaries. Appendix Example I v A ( σ | h ) with respect to σ A ( h ) and σ A ( h ) are ∂v A ( σ | h ) ∂σ A ( h ) = ∂v A ( σ | h ) ∂w A = w B ( w B + w B − w A + 2 w A + w B + 2 w B − w A + w B ) ( w A + w A + w B + w B − = 0 . (7.1) ∂v A ( σ | h ) ∂σ B ( h ) = ∂v A ( σ | h ) ∂w B = − ( w A ) ( w B + 2 ( w B − w A + w B ) ( w A + w A + w B + w B − − w A ( w A ( w B + 2 ( w B − w B ) )( w A + w B ) ( w A + w A + w B + w B − − w A (2 ( w B − w B + 2 ( w B − )( w A + w B ) ( w A + w A + w B + w B − − ( w A − w B ) ( w A + w B ) ( w A + w A + w B + w B − = 0 . (7.2)From equations (7 . 1) and (7 . 2) we conclude best response functions BR A and BR B suchthat BR A ( σ B ( h )) = 12 (cid:0) − w A − w B − w B + 200 (cid:1) ,BR B ( σ A ( h )) = − w A ( w A + w A + w B − w A + 2 ( w A − p w A ( w A + w A − w A + 2 ( w A − × p ( w A − w B + 100) ( w A + w A + w B − w A + 2 ( w A − . The intersection of best responses provides us the following two conditions w A = 12 (100 − w A ) ,w B = 12 (100 − w B ) . The first order conditions of v A ( σ ) with respect to σ A (ø) and σ A (ø) are ∂v A ( σ ) ∂w A = 2( w B − w A ) (3 w B − w A + w B − ( w A + w B ) + 2( w B − w A w B (3 w B − − w B − w B )( w A + w B − ( w A + w B ) = 0 , v A ( σ ) ∂w B = − w A − w A ) (3 w B − w A + w B − ( w A + w B ) − w A − w A (3( w B ) − w B + 20000)( w A + w B − ( w A + w B ) + 2( w A − w B ) )( w A + w B − ( w A + w B ) = 0 . From the above equations ∂v A ( σ ) /∂w A = 0 and ∂v A ( σ ) /∂w B = 0, we find the best responsefunctions of player A and B . 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