Sun Bins Legacy

A common format for sports contests involves pairwise matches between two teams, with the #1 player of team A matched against the #1 player of team B, the #2 player of team A against the #2 player of team B, and so on. This paper addresses the following question: What if team A wants to game the system? Can it gain an advantage by putting its players in a different order? The first example of this problem in history (to the author’s knowledge) was a match of three horse races between the King of Qi and general Tian Ji, in China, in the fourth century B.C. Tian Ji’s protege, Sun Bin, advised the general to throw the first race, running his #3 horse against the king’s #1, and then to race his #1 horse against the king’s #2 and his #2 horse against the king’s #3. In this way, Tian Ji won the match, 2-1, and Sun Bin ensured his reputation as a great strategist. The paper introduces a card-game version of this puzzle, called One Round War, and derives the optimal strategy in case each side has N cards (the analogue of N horses). The problem is recast as a linear assignment problem. The optimal solution involves sacrificing the weakest k cards/horses (for some k = k(N)) against the opponent’s strongest k in reverse order (weakest vs. strongest), and then playing the remaining N-k cards/horses against the opponent’s remaining cards in forward order. An exact formula is derived for k(N), and asymptotically we show that k(N) ~ sqrt (N ln N/2). Note: This paper proves that the strategy described above is optimal for sufficiently large N (e.g., for N greater than 10^7). It has also been verified by computer to be optimal for N less than 60. We conjecture that it is also true for N between 60 and 10^7. The upper bound 10^7 can be greatly improved, but closing the gap completely will probably require computer calculations that are beyond our resources.

💡 Research Summary

The paper revisits the ancient Chinese legend of Sun Bin’s horse‑racing trick and reformulates it as a modern combinatorial optimization problem called “One‑Round War.” In this game two players each receive N cards drawn from a deck numbered 1…2N. The cards are placed face‑down in order of strength (weakest to strongest) so that each player knows the relative ranking of his own hand and of the opponent’s hand, but not the exact values. Player 1 always leads a trick; player 2 must decide, before any cards are revealed, which of his cards to play against each of player 1’s cards. The goal for player 2 is to maximize the expected number of tricks won.

Mathematically the situation is captured by an N × N probability matrix P whose entry p_{ij} is the probability that player 2’s i‑th weakest card beats player 1’s j‑th weakest card. For a given permutation π (the strategy of player 2) the expected score is Σ_{i=1}^{N} p_{i,π(i)}. This is precisely a linear assignment problem: maximize the inner product ⟨M_π, P⟩ where M_π is the permutation matrix representing π. While the Hungarian algorithm can solve any such problem in polynomial time, the paper exploits the special structure of P to obtain a closed‑form optimal strategy.

The key observation is that P is a “mixed‑Monge matrix.” Mixed‑Monge matrices satisfy a Monge‑type inequality after possibly reversing rows or columns, and they are symmetric with respect to the anti‑main diagonal while being skew‑symmetric (up to a constant) about the main diagonal. These symmetries lead to two fundamental lemmas:

1. Symmetry Lemma – If a set of tricks is designated to be “thrown” (intentionally lost), the optimal way to throw them is to pair the i‑th weakest of those cards with the i‑th strongest opponent card, i.e. in reverse order.

2. No‑Gaps Theorem – For sufficiently large N (proved for N > 10⁷, conjectured true for all N ≥ 1), the set of thrown tricks forms a contiguous block {1,2,…,k}. In other words, there are no gaps between thrown and non‑thrown tricks in the optimal permutation.

Consequently the optimal permutation has the simple form

π_opt = (N, N‑1, …, N‑k+1, 1, 2, …, k),

meaning player 2 sacrifices his k weakest cards against the opponent’s k strongest cards (in reverse order) and then matches the remaining N‑k cards straight‑forwardly. The remaining question is to determine the optimal k = k*(N).

The paper discovers a striking connection between k*(N) and the central binomial coefficients. Define C(n, r) as the usual binomial coefficient. Let

S(k) = Σ_{j=0}^{N‑k} C(N‑1, j)².

The exact optimal k satisfies

k*(N) = max { k : S(k) ≥ ½ C(2N, N) }.

An almost identical “approximate” formula replaces the right‑hand side by C(2N, N) (without the factor ½); empirically the two values differ by at most one, and no counter‑example is known. The derivation uses Pascal’s triangle: the sum S(k) corresponds to the cumulative sum of squared entries in the (N‑1)‑st row, while ½ C(2N, N) is the central entry of the 2N‑th row. The paper provides worked examples for N = 7 and N = 13, showing how to locate the smallest k for which the inequality holds.

Asymptotically, Stirling’s approximation yields

k*(N) ∼ (N ln N)/2,

and the paper proves rigorous bounds

(N ln N)/4 < k*(N) < (N ln N)/2 for N > 400 (and verified by computer for 91 ≤ N ≤ 400). The convergence to the limit is extremely slow; even at N = 500 the ratio k*(N)/(N ln N) ≈ 0.559, far from the limiting 0.5.

The author also discusses a second, more “binary” objective: maximizing the probability of winning a simple majority of tricks, regardless of the total number of tricks won. For odd N = 2n + 1, Bill Courtney showed that the optimal strategy is to deliberately lose exactly n tricks, pairing the top n + 1 cards against the opponent’s bottom n + 1 cards in order. This guarantees the highest possible majority‑win probability but sacrifices expected trick count, losing roughly half the tricks on average. By contrast, the Pascal‑triangle strategy loses only about (N ln N)/2 tricks on average, allowing the player to approach a 100 % win rate as N grows.

The paper situates its contribution relative to earlier work. Assad, Golden, and Yee (1979) modeled a similar team‑matching problem but treated the probability matrix as arbitrary, offering only generic Hungarian‑algorithm solutions. The present work leverages the specific probabilistic structure of the card game to obtain an explicit optimal permutation. Another line of research (David & Liu) examined “fair matchings” that equalize expected wins; here the focus is on the opposite – the most “unfair” matching that maximizes one side’s advantage.

The remainder of the paper is organized as follows: Section I introduces the linear‑assignment formulation, defines mixed‑Monge matrices, and proves a weak shape theorem that any optimal strategy must consist of a reverse‑ordered block and a forward‑ordered block (possibly with gaps). Section II proves the Symmetry Lemma using the anti‑diagonal symmetry of the matrix. Section III contains the technical No‑Gaps Theorem, showing that for large N any gap would be suboptimal. Section IV derives the exact and approximate formulas for k*(N) via Pascal’s triangle, establishes the asymptotic behavior, and presents numerical tables for small N. The paper concludes with remarks on the majority‑win problem, open questions about extending the proof to moderate N, and potential applications of mixed‑Monge structures to other assignment problems.

In summary, the study transforms a centuries‑old tactical anecdote into a rigorous combinatorial optimization result. By identifying the mixed‑Monge nature of the underlying probability matrix and exploiting its symmetries, the author obtains a simple closed‑form optimal strategy: sacrifice a contiguous block of the weakest cards against the opponent’s strongest, with the block size given by a precise binomial‑coefficient inequality. This not only settles Sun Bin’s ancient puzzle for arbitrary N but also enriches the theory of linear assignment problems with a new class of analytically tractable instances.