Blotto games with costly winnings

Abstract

We introduce a new variation of the m-player asymmetric Colonel Blotto game, where the n battles 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.

Full text

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
nck+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 nck+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+b1w1+(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 xW 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