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BLOTTO GAMES WITH COSTLY WINNINGS

IRIT NOWIK AND TAHL NOWIK

Abstract. We introduce a new variation of the m-player asymmetric Colonel Blotto

game, where the nbattles occur as sequential stages of the game, and the winner of

each stage needs to spend resources for maintaining his win. The limited resources

of the players are thus needed both for increasing the probability of winning and for

the maintenance costs. We show that if the initial resources of the players are not

too small, then the game has a unique Nash equilibrium, and the given equilibrium

strategies guarantee the given expected payoﬀ for each player.

1. Introduction

We present a new n-stage game, which is a variation of the Colonel Blotto game.

Each player starts the game with some given resource, and at the beginning of each

stage he must decide how much resource to invest in that stage. A player wins the

given stage with probability corresponding to the relative investments of the players,

and if all players invest 0 then no player wins that stage. The winner of the stage

receives a payoﬀ which may diﬀer from stage to stage. Since it is possible that certain

stages will not be won by any player, this is not a ﬁxed sum game.

The players’ resources from which the investments are taken can be thought of as

money, whereas the payoﬀs should be thought of as a quantity of diﬀerent nature, such

as political gain. The two quantities cannot be interchanged, that is, the payoﬀ cannot

be converted into resources for further investment.

The new feature of our game is the following. The winner of each stage is required

to spend additional resources on the maintenance of his winning. This is a real life

situation, where the winnings are some assets, and resources are required for their

maintenance, as in wars, territorial contests among organisms, or in the political arena.

The winner of a given stage must put aside all resources that will be required for future

maintenance costs of the won asset. Thus, a ﬁxed amount will be deducted from the

Date: March 3, 2016.

1

2 IRIT NOWIK AND TAHL NOWIK

resources of the winner immediately after winning, which should be thought of as the

sum of all future maintenance costs for the given acquired asset.

At each stage the player thus needs to decide how much to invest in the given

stage, where winning that stage on one hand leads to the payoﬀ of the given stage,

but on the other hand the maintenance cost for the given winning negatively aﬀects

the probabilities for future winnings. In the present work we show that if the initial

resources of the players are not too small then the game has a unique Nash equilibrium,

and each player guarantees the payoﬀ of this Nash equilibrium (Theorem 3.5 for two

players and Theorem 4.1 for mplayers.)

As mentioned, our game presents a variation of the well known Colonel Blotto game

([B]). In Blotto games two players simultaneously distribute forces across several bat-

tleﬁelds. At each battleﬁeld, the player that allocates the largest force wins. The

Blotto game has been developed and generalized in many directions (see e.g., [B], [F],

[L], [R], [H], [DM]). Two main developments are the “asymmetric” and the “stochas-

tic” models. The asymmetric version allows the payoﬀs of the battleﬁelds to diﬀer from

each other, and in the stochastic model the deterministic rule deciding on the winner is

replaced by a probabilistic one, by which the chances of winning a battleﬁeld depends

on the size of investment.

The present work adds a new feature which changes the nature of the game, in

making the winnings costly. The players thus do not know before hand how much of

their resources will be available for investing in winning rather than on maintenance,

and so the game cannot be formulated with simultaneous investments, as in the usual

Blotto games, but rather must be formulated with sequential stages. At each stage the

players need to decide how much to invest in the given stage, based on their remaining

available resources and on the future fees and payoﬀs.

This work was inspired by previous work of the ﬁrst author with S. Zamir and I.

Segev ([N],[NZS]) on a developmental competition that occurs in the nervous system,

which we now describe. A muscle is composed of many muscle-ﬁbers. At birth each

muscle-ﬁber is innervated by several motor-neurons (MNs) that “compete” to singly

innervate it. It has been found that MNs with higher activation-threshold win in

more competitions than MNs with lower activation thresholds. In [N] this competitive

BLOTTO GAMES WITH COSTLY WINNINGS 3

process is modeled as a multi-stage game between two groups of players: those with

lower and those with higher thresholds. At each stage a competition at the most active

muscle-ﬁber is resolved. The strategy of a group is deﬁned as the average activity level

of its members and the payoﬀ is deﬁned as the sum of their wins. If a MN wins (i.e.,

singly innervates) a muscle-ﬁber, then from that stage on, it must continually devote

resources for maintaining this muscle-ﬁber. Hence the MNs use their resources both

for winning competitions and for maintaining previously acquired muscle-ﬁbers. It is

proved in [N] that in such circumstances it is advantageous to win in later competitions

rather than in earlier ones, since winning at a late stage will encounter less maintenance

and thus will negatively aﬀect only the few competitions that were not yet resolved. If

µis the cost of maintaining a win at each subsequent stage, then in the terminology

of the present work, the fee payed by the MNs for winning the kth stage of an nstage

game is (n−k)µ.

We start by analyzing the 2-person game in Sections 2 and 3, and then generalize to

the m-person setting in Section 4.

2. The 2-person game

The initial data for the 2-person version of our game is the following.

(1) The number nof stages of the game.

(2) Fixed payoﬀs wk>0, 1 ≤k≤n, to be received by the winner of the kth stage.

(3) The initial resources A, B ≥0 of players I,II respectively.

(4) Fixed fees ck≥0, 1 ≤k≤n−1, to be deducted from the resources of the

winner after the kth stage.

The rules of the game are as follows. At the kth stage of the game, the two players,

which we name PI,PII, each has some remaining resource Ak, Bk, where A1=A, B1=

B. PI,PII each needs to decide his investment xk, ykfor that stage, respectively, with

0≤xk≤Ak−ck, 0 ≤yk≤Bk−ck, and where if Ak< ckthen PI may only invest 0,

and similarly for PII. These rules ensure that the winner of the given stage will have

the resources for paying the given fee ck. The probability for PI,PII of winning this

stage is respectively xk

xk+ykand yk

xk+yk, where if xk=yk= 0 then no player wins. The

resource of the winner of the kth stage is then reduced by an additional ck, that is,

4 IRIT NOWIK AND TAHL NOWIK

if PI wins the kth stage then Ak+1 =Ak−xk−ckand Bk+1 =Bk−yk, and if PII

wins then Ak+1 =Ak−xkand Bk+1 =Bk−yk−ck. The role of ckis in determining

Ak+1, Bk+1, thus there is no cn. It will however be convenient in the sequel to formally

deﬁne cn= 0. The payoﬀ received by the winner of the kth stage is wk. Since it is

possible that no player wins certain stages, this game is not a ﬁxed sum game.

As already mentioned, the resource quantities Ak, Bk, xk, yk, ckused for the invest-

ments and fees are of diﬀerent nature than that of the payoﬀs wk. The two quantities

cannot be interchanged and should be thought of as having diﬀerent “units”. Note

that all expressions below are unit consistent, that is, if say we divide resources by

payoﬀ, then such expression has units of resources

payoﬀ , and may only be added or equated

to expressions of the same units.

If Aand Bare too small in comparison to c1, . . . , cn−1then the players’ strategies are

strongly inﬂuenced by the possibility of running out of resources before the end of the

game. In the present work we analyze the game when A, B are not too small. Namely,

we introduce a quantity Mdepending on c1, . . . , cn−1and w1, . . . , wn, and prove that

if A, B > M then there is a unique Nash equilibrium for the game, and each player

guarantees the value of this Nash equilibrium.

For k= 1, . . . , n let Wk=Pn

i=kwiand W=W1. We now show that if A>M, then

if PI always chooses to invest xk≤wk

WkAk(as holds for our strategy σn,A,B presented

in Deﬁnition 3.1 below), then whatever the random outcomes of the game are, his

resources will not run out before the end of the game. We in fact give a speciﬁc lower

bound on Akfor every k, which will be used repeatedly in the sequel.

Proposition 2.1. Let

M=W·max

1≤k≤n ck

wk

+

k−1

X

i=1

ci

Wi+1 !.

If A>M, and if PI plays xk≤wkAk

Wkfor all k, then Ak>Wkck

wkfor all 1≤k≤n. In

particular Ak>0for all 1≤k≤n. And similarly for PII.

Proof. For every 1 ≤k≤nwe have A

W>M

W≥ck

wk+Pk−1

i=1

ci

Wi+1 , so

A

W−

k−1

X

i=1

ci

Wi+1

>ck

wk

.

BLOTTO GAMES WITH COSTLY WINNINGS 5

Thus it is enough to show that Ak

Wk≥A

W−Pk−1

i=1

ci

Wi+1 for all 1 ≤k≤n. We show this

by induction on k. For k= 1 the sum is empty and we get equality. Assuming

Ak

Wk

≥A

W−

k−1

X

i=1

ci

Wi+1

we get

Ak+1

Wk+1

≥1

Wk+1 Ak−wkAk

Wk

−ck=1

Wk+1 Wk+1Ak

Wk

−ck

=Ak

Wk

−ck

Wk+1

≥A

W−

k−1

X

i=1

ci

Wi+1

−ck

Wk+1

=A

W−

k

X

i=1

ci

Wi+1

.

We consider two simple examples of c1, . . . , cn−1,w1, . . . , wn, for which Mmay be

readily identiﬁed.

(1) Let ck=n−k,wk= 1 for all k. These fees and payoﬀs are as in the

biological game described in the introduction. For every 1 ≤k≤nwe have

nck+Pk−1

i=1

ci

n−i=n(n−k)+(k−1)=n(n−1), so M=n(n−1).

(2) Let ck= 1 for all 1 ≤k≤n−1, wk= 1 for all k. Using the inequality Pn

k=1

1

k<

1 + ln n, we get for every 1 ≤k≤nthat nck+Pk−1

i=1

ci

n−i< n(2 + ln n), so

M < n(2 + ln n).

We note that an obvious necessary condition for Ato satisfy Proposition 2.1 is

A≥Pn−1

k=1 ck, since in case PI wins all stages he will need to pay all fees ck. We see

that Min the two examples above is not much larger. Namely, in (1), Pn−1

k=1 ck=n(n−1)

2

and M=n(n−1), and in (2), Pn−1

k=1 ck=n−1 and M < n(2 + ln n).

3. Nash equilibrium

We deﬁne the following two strategies σn,A,B and τn,A,B for PI,PII respectively. We

prove that for A, B > M as given in Proposition 2.1, this pair of strategies is a unique

Nash equilibrium, and these strategies guarantee the given payoﬀs.

Deﬁnition 3.1. At the kth stage of the game, let

ak=wkAk

Wk

−Akck

Ak+Bk

and bk=wkBk

Wk

−Bkck

Ak+Bk

.

6 IRIT NOWIK AND TAHL NOWIK

where as mentioned, we formally deﬁne cn= 0. The strategy σn,A,B for PI is the

following: At the kth stage PI invests akif it is allowed by the rules of the game.

Otherwise he invest 0. The strategy τn,A,B for PII is similarly deﬁned with bk.

Recall that ak6= 0 is allowed by the rules of the game if 0 ≤ak≤Ak−ck, whereas

ak= 0 is always allowed, even when Ak−ck<0. We interpret the quantities ak, bkas

follows. PI ﬁrst divides his remaining resource Akto the remaining stages in proportion

to the payoﬀ for each remaining stage, which gives wk

WkAk. From this he subtracts

Ak

Ak+Bkckwhich is the expected fee he will pay for this stage, since ak

ak+bk=Ak

Ak+Bk. Note

that Wn=wnand formally cn= 0, so an=An,bn=Bn, i.e. at the last stage the

two players invest all their remaining resources.

Depending on Aand Band on the random outcomes of the game, it may be that PI

indeed reaches a stage where akis not allowed. In this regard we make the following

deﬁnition.

Deﬁnition 3.2. The triple (n, A, B) is PI-eﬀective if when PI and PII use σn,A,B and

τn,A,B, then it is impossible that they reach a stage where akis not allowed for PI.

Similarly PII-eﬀectiveness is deﬁned for PII with bk.

Proposition 3.3. Let Mbe as in Proposition 2.1. If A>M and Bis arbitrary, then

(n, A, B)is PI-eﬀective. Furthermore, ak>0for all k. And similarly for PII when

B > M .

Proof. We need to show that necessarily 0 < ak≤Ak−ckfor all 1 ≤k≤n. We have

ak=wkAk

Wk−Akck

Ak+Bk≤wkAk

Wk, so by Proposition 2.1, wk

Wk>ck

Ak≥ck

Ak+Bkand Ak>0, so

wkAk

Wk>Akck

Ak+Bkgiving ak>0.

For the inequality ak≤Ak−ckwe ﬁrst consider k≤n−1. We have from the proof

of Proposition 2.1 that Ak

Wk−ck

Wk+1 ≥A

W−Pk

i=1

ci

Wi+1 >ck+1

wk+1 ≥0, so Ak

Wk>ck

Wk+1 , and so

(1 −Ak

Ak+Bk

)ck≤ck<Wk+1Ak

Wk

= (1 −wk

Wk

)Ak.

This gives ck−Akck

Ak+Bk< Ak−wkAk

Wk, so ak=wkAk

Wk−Akck

Ak+Bk< Ak−ck. For k=nwe

note that cn= 0 by deﬁnition, and Wn=wn, so an=An=An−cn.

In general, an inductive characterization of PI-eﬀectiveness will also involve induction

regrading PII. But if we assume that B > M , and so by Proposition 3.3 all bkare known

BLOTTO GAMES WITH COSTLY WINNINGS 7

to be allowed and positive, then the notion of PI-eﬀectiveness becomes simpler, and

may be characterized inductively as follows. When saying that a triple (n−1, A0, B0)

is PI-eﬀective, we refer to the n−1 stage game with fees c2, . . . , cn−1and payoﬀs

w2, . . . , wn. Starting with n= 1, (1, A, B ) is always PI-eﬀective. For n≥2, if a1

is not allowed then (n, A, B) is not PI-eﬀective. If a1= 0 then it is allowed, and PI

surely loses the ﬁrst stage, and so (n, A, B) is PI-eﬀective iﬀ (n−1, A, B −b1−c1) is

PI-eﬀective. Finally if a1>0 and it is allowed then (n, A, B) is PI-eﬀective iﬀ both

(n−1, A −a1−c1, B −b1) and (n−1, A −a1, B −b1−c1) are PI-eﬀective.

The crucial step in proving Theorem 3.5 below, on the unique Nash equilibrium and

the guaranteed payoﬀs, is the following Theorem 3.4. We point out that in Theorem 3.5

we will assume that A>M, in which case (n, A, B) is PI-eﬀective, by Proposition 3.3.

But here in Theorem 3.4 we must consider arbitrary A≥0 in order for an induction

argument to carry through.

Theorem 3.4. Given c1, . . . , cn−1and w1, . . . , wnlet Mbe as in Proposition 2.1, and

assume that B > M and PII plays the strategy τn,A,B. For A≥0, if (n, A, B)is

PI-eﬀective, and PI plays according to σn,A,B, then his expected payoﬀ is AW

A+B. On the

other hand, if (n, A, B)is not PI-eﬀective, or if PI uses a diﬀerent strategy, then his

expected payoﬀ is strictly less than AW

A+B.

Proof. By induction on n. We note that throughout the present proof we do not use the

condition B > M directly, but rather only through the statements of Propositions 3.3

and 2.1 saying that (n, A, B) is PII-eﬀective, bk>0 and ck<wkBk

Wkfor all 1 ≤k≤n,

which indeed continue to hold along the induction process.

If A= 0 then ak= 0 for all k, which is the only possible investment, and its payoﬀ

is 0 = AW

A+B, so the statement holds. We thus assume from now on that A > 0. For

n= 1 we have b1=B. The allowed investment for PI is 0 ≤s≤Awith expected

payoﬀ s

s+Bw1=s

s+BWwhich indeed attains a strict maximum A

A+BWat s=A=a1.

For n≥2, let sbe the investment of PI in the ﬁrst stage. Assume ﬁrst that

s= 0. In this case PII surely wins the ﬁrst stage and so following this stage we have

A2=Aand B2=B−b1−c1. The moves for PII dictated by τn,A,B for the remaining

n−1 stages of the game are τn−1,A,B−b1−c1, and so by the induction hypothesis the

8 IRIT NOWIK AND TAHL NOWIK

expected total payoﬀ of PI is at most AW2

A+B−b1−c1. Since Proposition 2.1 holds for

PII, we have c1<w1B

W≤w1(A+B)

W, that is, w1

W−c1

A+B>0, and since A > 0 we

get a1=A(w1

W−c1

A+B)>0. This means that s= 0 6=a1, so we must verify the

strict inequality AW2

A+B−b1−c1<AW

A+B. This is readily veriﬁed, using A > 0, c1<w1B

W,

W2=W−w1, and b1+c1=w1B

W−Bc1

A+B+c1=w1B

W+Ac1

A+B.

We now assume s > 0. This is allowed only if A>c1and 0 < s ≤A−c1. The moves

for PII dictated by τn,A,B for the remaining n−1 stages of the game are τn−1,A2,B2.

By the induction hypothesis, if PI wins the ﬁrst stage, which happens with probability

s

s+b1>0, then his expected payoﬀ in the remaining n−1 stages of the game is at

most (A−s−c1)W2

A+B−s−b1−c1. Similarly, if he loses the ﬁrst stage, which happens with probability

b1

s+b1>0, then his expected payoﬀ in the remaining n−1 stages is at most (A−s)W2

A+B−s−b1−c1.

Thus, the expected payoﬀ of PI for the whole nstage game is at most F(s), where

F(s) = s

s+b1w1+(A−s−c1)W2

A+B−s−b1−c1+b1

s+b1

·(A−s)W2

A+B−s−b1−c1

with b1=w1B

W−Bc1

A+B.

By the induction hypothesis we know furthermore, that in case PI wins the ﬁrst stage,

he will attain the maximal expected payoﬀ (A−s−c1)W2

A+B−s−b1−c1in the remaining stages of the

game only if (n−1, A −s−c1, B −b1) is PI-eﬀective, and he uses σn−1,A−s−c1,B−b1. Sim-

ilarly, if he loses the ﬁrst stage, he will attain the maximal expected payoﬀ (A−s)W2

A+B−s−b1−c1

only if (n−1, A −s, B −b1−c1) is PI-eﬀective and he uses σn−1,A−s,B−b1−c1. If not,

then since both alternatives occur with positive probability, his expected total payoﬀ

for the whole nstage game will be strictly less than F(s).

To analyze F(s), we make a change of variable s=a1+x, that is, we deﬁne

b

F(x) = F(a1+x) = F(w1A

W−Ac1

A+B+x). After some manipulations we get:

b

F(x) = AW

A+B−BW 3x2

(A+B)W2(A+B)−W xW x −W c1+w1(A+B).

Under this substitution, s=a1corresponds to x= 0, and the allowed domain

0< s ≤A−c1corresponds to

Ac1

A+B−w1A

W< x ≤W2A

W−Bc1

A+B.

BLOTTO GAMES WITH COSTLY WINNINGS 9

Using c1<w1B

W, one may verify that in the above expression for b

Fthe two linear

factors appearing in the denominator of the second term are both strictly positive in

this domain. It follows that b

Fin the given domain is at most AW

A+B, and this maximal

value is attained only for x= 0 (if it is in the domain), which corresponds to s=a1

for the original F. Finally, as mentioned, unless (n−1, A2, B2) is PI-eﬀective and PI

plays σn−1,A2,B2, his expected payoﬀ will be strictly less than F(s), and so strictly less

than AW

A+B.

We may now prove our main result.

Theorem 3.5. Given c1, . . . , cn−1and w1, . . . , wn, let Mbe as in Proposition 2.1,

and assume A, B > M . Then the pair of strategies σn,A,B, τn,A,B is a unique Nash

equilibrium for the game, with expected total payoﬀs AW

A+B,BW

A+B. Furthermore, σn,A,B

and σn,A,B guarantee the expected payoﬀs AW

A+Band BW

A+B.

Proof. Denote σ0=σn,A,B and τ0=τn,A,B, and for any pair of strategies σ, τ let

S1(σ, τ), S2(σ, τ ) be the expected payoﬀs of PI, PII respectively. We ﬁrst prove the

second statement of the theorem. Recall that if both players invest 0 in a given stage

then there is no winner to that stage. However, if B > M and PII plays τ0, then

by Proposition 3.3 we have bk>0 for all k, and so indeed there is a winner to each

stage of the game, and thus the total combined payoﬀ of PI and PII is necessarily W.

It thus follows from Theorem 3.4 that for any strategy σof PI we have S2(σ, τ0) =

W−S1(σ, τ0)≥BW

A+B. Similarly, if A>M then S1(σ0, τ)≥AW

A+Bfor all τ, establishing

the second statement of the theorem.

As to the ﬁrst statement, Theorem 3.4 applied to both PI and PII implies that the

pair σ0, τ0is a Nash equilibrium with the given expected payoﬀs. To show it is unique

we argue as follows. Let σ, τ be any other Nash equilibrium and assume that σ6=σ0. By

Theorem 3.4 we have S1(σ, τ0)< S1(σ0, τ0) and since playing τ0guarantees a combined

total payoﬀ of W, we have S2(σ, τ0) = W−S1(σ, τ0)> W −S1(σ0, τ0) = S2(σ0, τ0).

Since the pair σ, τ is a Nash equilibrium we also have S2(σ, τ)≥S2(σ, τ0), and together

we get S2(σ, τ)> S2(σ0, τ0). Since S1(σ, τ )+ S2(σ, τ)≤Wand S1(σ0, τ0) + S2(σ0, τ0) =

W, we must have S1(σ, τ )< S1(σ0, τ0). Again since σ, τ is a Nash equilibrium we

10 IRIT NOWIK AND TAHL NOWIK

have S1(σ0, τ)≤S1(σ, τ ) so together S1(σ0, τ)< S1(σ0, τ0) = AW

A+B, contradicting the

conclusion of the previous paragraph.

We remark that our Nash equilibrium is subgame perfect. This may be seen from

the inductive proof, and also follows from the uniqueness of the Nash equilibrium, since

the game is ﬁnite.

4. The m-person game

The generalization of our game and results to an m-person setting is straightforward.

Player Pi, 1 ≤i≤m, starts with resource Ai≥0, and we are again given ﬁxed

fees c1, . . . , cn−1≥0, and payoﬀs w1, . . . , wn>0. At stage 1 ≤k≤nplayer Pi

has remaining resource Ai

k, with Ai

1=Ai. At stage keach player decides to invest

0≤xi

k≤Ai

k−ck, or 0 if Ai

k< ck. The probability for Pito win is xi

k

Pm

j=1 xj

k

, and if

xi

k= 0 for all mthen no player wins. The winner of the kth stage receives payoﬀ wk

and pays the maintenance fee ck.

We deﬁne Mas before, M=W·max1≤k≤n ck

wk+Pk−1

i=1

ci

Wi+1 !.

The strategy σi

n,A1,...,Amof Piis the straightforward generalization of the strategies

σn,A,B, τn,A,B of the 2-person game, namely, at the kth stage Piinvests

ai

k=wkAi

k

Wk

−Ai

kck

Pm

j=1 Aj

k

if it is allowed, and 0 otherwise.

The generalization of Theorem 3.5 is the following.

Theorem 4.1. If A1, . . . , Am> M then the mstrategies σ1

n,A1,...,Am, . . . , σm

n,A1,...,Am

are a unique Nash equilibrium for the m-person game, with expected total payoﬀs

A1W

Pm

i=1 Ai,..., AmW

Pm

i=1 Ai. Furthermore, σi

n,A1,...,Amguarantees the expected payoﬀ AiW

Pm

j=1 Ajfor

Pi.

Proof. We prove for P1. Let Bk=Pm

i=2 Ai

kand bk=Pm

i=2 ai

k. Since a1

k=wkA1

k

Wk−A1

kck

A1

k+Bk

and bk=wkBk

Wk−Bkck

A1

k+Bk, our player P1can imagine that he is playing a 2-person game

against one joint player whose resource is Bk, and whose strategy dictates investing bk.

Thus the claim follows from Theorem 3.5.

BLOTTO GAMES WITH COSTLY WINNINGS 11

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Department of Industrial Engineering and Management, Lev Academic Center,

P.O.B 16031, Jerusalem 9116001, Israel

E-mail address:nowik@jct.ac.il

Department of Mathematics, Bar-Ilan University, Ramat-Gan 5290002, Israel

E-mail address:tahl@math.biu.ac.il

URL:www.math.biu.ac.il/~tahl