Measured Games  ·  Paper 06 of 07  ·  Operations research

Colonel Blotto

After É. Borel (1921); solved in general by B. Roberson (2006)

Abstract

Two players allocate fixed budgets across several fronts, each front falling to the higher bid. No pure strategy survives, since any allocation can be countered by conceding its strongest fronts and narrowly outbidding it elsewhere. Equilibrium is a distribution over allocations with uniform marginals on the support, and asymmetric budgets produce an atom at zero for the weaker player: the underdog abandons fronts rather than spreading thin.

Keywordsresource allocation · mixed equilibrium · asymmetric budgets · all-pay contest

1Setup

Setup

Two players each hold a fixed, non-transferable budget. Both simultaneously divide it across n fronts. Each front goes to whoever committed more there. The payoff is the number of fronts won. Nothing is refunded, and beating an opponent by a wide margin on a front scores exactly the same as beating them by a hair.

The reason no fixed plan can work is short. Whatever you commit, an opponent who knows it can concede your heaviest fronts at zero cost and redirect that budget to narrowly outbid you everywhere else. Since that holds for every allocation, none of them can be an equilibrium, and the solution has to be a distribution over allocations rather than a plan.

2The model

The model

# budgets X and Y across n fronts Σ xi ≤ X Σ yi ≤ Y front i to A if xi > yi # Roberson (2006), symmetric budgets, continuous case equilibrium marginals are uniform on [ 0 , 2X/n ] every front is contested with the same law # asymmetric budgets, X < Y the weaker player abandons fronts with positive probability and concentrates the remainder

Which fronts the weaker player abandons has to stay random. A predictable concentration is countered by conceding exactly those fronts, which is the same argument that ruled out pure strategies in the first place.

Figure 1Allocation against a Roberson equilibrium opponent

Current round · your allocation above, opponent below

Running fronts won per round, against the theoretical equilibrium payoff

Your empirical marginal against the equilibrium law

You Opponent Equilibrium law Running average

Charts are focusable. With a chart focused, use the left and right arrow keys to read values along the horizontal axis.

Fronts won per round—
Rounds won—
Budget ratio—
Equilibrium payoff—
Fronts abandoned—
Rounds played0
Figure 1. The opponent draws from the Roberson equilibrium by rejection sampling, so its marginals really are uniform on the equilibrium support and really do sum to its budget. The equilibrium payoff readout is the theoretical value, n·X/2Y for the weaker side, and your measured result should sit on it when you switch to the equilibrium draw. Two things to notice: an even spread underperforms the equilibrium at unequal budgets, and concentration raises expected fronts won while making an outright majority impossible once you hold fewer than half the fronts.The middle panel traces your running fronts-won average against the theoretical equilibrium payoff, so a strategy that is quietly losing shows up as a curve settling below the line rather than on it.
3Results

Results

Borel (1921) posed the problem and it resisted general solution for eighty-five years. Gross and Wagner (1950), working at RAND, solved specific symmetric cases. Roberson (2006) gave the complete characterisation of equilibrium for the continuous game with arbitrary, possibly asymmetric budgets, along with the equilibrium payoffs.

The strategic content lives in the asymmetric case. A player with a smaller budget cannot win an even spread and should not attempt one. The equilibrium has the weaker player abandoning some fronts entirely with positive probability and concentrating the remainder, while the stronger player spreads more evenly to cover everything. Hart (2008) worked out the discrete version, where budgets are integers and ties are possible, which behaves differently in ways that matter for any application with indivisible resources.

4Evidence

Evidence

Chowdhury, Kovenock and Sheremeta (2013) ran the constant-sum game with asymmetric budgets in a laboratory, with subjects maximising the expected number of fronts won. They compared two ways of resolving a front. In the auction treatment the higher allocation wins outright, which is the game as stated above. In the lottery treatment the higher allocation only wins with higher probability.

The results support the main qualitative predictions, and the auction treatment reproduces the asymmetric result directly. Disadvantaged players adopted what the authors call a guerilla warfare strategy: stochastically allocating zero to a subset of fronts and concentrating the rest. Advantaged players used stochastic complete coverage, allocating random but everywhere-positive amounts. That is the atom at zero for the weaker side and the spread for the stronger side, observed in human subjects rather than derived. It is also exactly what the abandoned-fronts readout in the figure above counts.

The lottery treatment behaves differently, as theory expects: when winning is probabilistic rather than certain, both sides simply divide their budgets evenly across all fronts. The authors also report behavioural deviations from the theory alongside the confirmations.

Outside the laboratory the model is the standard formalisation of campaign resource allocation across states under the electoral college, and the same structure is used for research and development portfolios, multi-item auctions, and network security where a defender allocates protection across targets an attacker will probe.

5Limitations

Limitations

The empirical status here is different in kind from the biological sheets and should be stated plainly. There is good laboratory evidence that people approximate the qualitative predictions, including the distinctive underdog behaviour, and there is widespread applied use of the framework. There is nothing comparable to a field census: these are subjects in an experiment, not colonels.

The model also assumes both players know both budgets, that fronts are equally valuable unless weighted by hand, that allocations are simultaneous and that winning a front by any margin is worth the same. Campaign spending violates most of these: budgets are uncertain and dynamic, states differ enormously in value, spending is sequential and observable, and margins matter for downstream races. The framework is useful for the shape of the answer, which is that underdogs concentrate and favourites spread, rather than for the numbers.

Related This model shares its mechanism with Cyclic dominance (Paper 02) and The war of attrition (Paper 05) and Professionals play minimax (Paper 08). In each of them, no fixed choice is stable, so the solution is a distribution or a cycle rather than a move.

 References

References

  1. Borel, E. (1953). The theory of play and integral equations with skew symmetric kernels (L. J. Savage, Trans.). Econometrica, 21(1), 97–100. (Original work published 1921) https://doi.org/10.2307/1906946
  2. Chowdhury, S. M., Kovenock, D., & Sheremeta, R. M. (2013). An experimental investigation of Colonel Blotto games. Economic Theory, 52(3), 833–861. https://doi.org/10.1007/s00199-011-0670-2
  3. Gross, O. A., & Wagner, R. A. (1950). A continuous Colonel Blotto game (Research Memorandum RM-408). RAND Corporation. https://www.rand.org/pubs/research_memoranda/RM408.html
  4. Hart, S. (2008). Discrete Colonel Blotto and General Lotto games. International Journal of Game Theory, 36(3–4), 441–460. https://doi.org/10.1007/s00182-007-0099-9
  5. Kovenock, D., & Roberson, B. (2012). Coalitional Colonel Blotto games with application to the economics of alliances. Journal of Public Economic Theory, 14(4), 653–676. https://doi.org/10.1111/j.1467-9779.2012.01556.x
  6. Roberson, B. (2006). The Colonel Blotto game. Economic Theory, 29(1), 1–24. https://doi.org/10.1007/s00199-005-0071-5

Verification Every reference on this page has been checked against Crossref. The claims this page makes about what those sources report have also been checked against the sources, and any corrections that check produced are recorded in the project’s claims log.