Engine · Chapter 2, Competing for Allocation: The Blotto Game

Blotto

How should a fixed budget be spread across contested fields against an adversary?

Budgets

A's units 6

B's units 6

Field weights

Field 1 1

Field 2 1

Field 3 1

Ties

split the field go to B

The book's scenarios

Value of the game to A

0

A's weighted score minus $W /2 = 3/2$ : positive means A wins more than half the weight in expectation against B's best reply, whatever B does.

D Base: 6 vs 6, w=(1,1,1), tie split: the book certifies value $0$ ; its strategies, evaluated here, guarantee $0$ and expose $0$ . Agreement, exactly.

A's optimal mixture: 7 allocations of 6

B's optimal mixture: 3 allocations of 6

What a strategy of A guarantees

Strategy	Guarantee $min_{y} p^{⊤} G$
Optimal mixture	$0$	7 allocations in support
Uniform over all allocations	$- \frac{9}{56}$	28 allocations, equally likely
Pure (2, 2, 2)	$- \frac{1}{2}$	split evenly, no randomisation
Permutations of (0, 2, 4)	$0$	abandon one field, equally likely which

Pick an allocation on A's triangle to see what it guarantees on its own.

The certificate:

L = 0 = U

in exact arithmetic, 68 pivots

A's mixture $p^{*}$

(0, 3, 3): 1/7
(1, 1, 4): 1/7
(1, 4, 1): 1/7
(2, 1, 3): 1/7
(3, 2, 1): 1/7
(3, 3, 0): 1/7
(4, 0, 2): 1/7

$L = min_{j} (p^{* ⊤} G)_{j} = 0$

B's mixture $q^{*}$

(0, 2, 4): 1/3
(2, 4, 0): 1/3
(4, 0, 2): 1/3

$U = max_{i} (G q^{*})_{i} = 0$

Since $L \leq v \leq U$ for every pair of mixtures and here $L = U$ , both mixtures are optimal and the value is $0$ . The payoff matrix is 28 × 28; the linear programme was solved by an exact simplex in your browser, so the fractions above are the solution, not a rounding of it.

The model it runs

base allocation game

What can each side guarantee, and which allocations does every optimal strategy avoid?

1Decision-makers: two, $A$ (rows) and $B$ (columns)
2Time: one shot
3Information: complete; neither side sees the other's allocation before choosing (Section 2.10.4 (p. 97) reveals one field)
4Commitment: $Γ_{0}$ ; announcing a mixture is a perturbation, free here (Section 0.4 (p. 10))
5Payoffs: zero-sum after centring: $g = u - W /2$ , $u$ = weight won, ties counted half
6Strategies: mixtures $p \in Δ (X_{A})$ , $q \in Δ (X_{B})$ over integer allocations, drawn independently

Analysis contract

(i) concept: value and optimal strategies of a finite zero-sum game (thm:ch00-minimax,thm:ch00-bridge)
(ii) deviations: every unilateral deviation to a pure allocation; no further consistency condition
(iii) beliefs: none needed: complete information, one simultaneous move
(iv) selection: none for the value, which is unique; claims about strategies separate existence (one certified example) from necessity (every optimal strategy)
(v) attainment: attained; exact rational certificate $L = U$ ; necessity claims by coordinate-LP dual witnesses