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Silverman's game

A two-player zero-sum number-selection game that rewards a moderately larger choice but penalizes a choice whose ratio to the opponent's reaches a fixed threshold.

Version
v1 · 2026-09-28 · History
Domain-specific #
12048
Domain group
Formal Sciences
Origin domain
Mathematics
Subdomains
Game Theory, Zero Sum Games → Mathematics

Core Idea

Silverman's game asks two players to choose positive numbers independently from declared sets. If their choices tie, the payoff is zero; if one is larger but remains below a threshold multiple of the other, that player wins.

The distinctive twist is overshoot: when the larger-to-smaller ratio reaches the threshold, the larger chooser incurs a fixed penalty. Strategy sets may be finite, countable, continuous, or player-specific, but the thresholded ratio payoff remains constitutive.

Structural Signature

Sig role-phrases:

  • Strategy sets — Specify the admissible positive numbers for each player. It is carrier. Counterfactual: Changing the sets changes the game instance but need not change the payoff rule.
  • Threshold T — Separates a rewarded larger choice from an overlarge penalized choice. It is parameter. Counterfactual: Without T greater than one, the characteristic middle interval disappears.
  • Penalty nu — Sets the loss for exceeding the ratio threshold. It is parameter. Counterfactual: A zero or unspecified penalty changes strategic incentives.
  • Simultaneous choices — Prevent either player from conditioning a choice on the other's realized number. It is protocol. Counterfactual: Sequential observation defines another game.
  • Ratio comparison — Normalizes how much larger one selected number is than the other. It is operation. Counterfactual: Absolute difference is not the stated payoff basis.
  • Zero-sum payoff — Assigns opposite outcomes to the two players for every choice pair. It is semantics. Counterfactual: A shared reward would not be Silverman's game.

What It Is Not

  • It is not a generic largest-number-wins contest.
  • It is not an auction or bidding mechanism.
  • It is not defined by absolute numerical difference.
  • It is not every two-player zero-sum game on a square.
  • Closest near-miss. A largest-number-wins game rewards every strictly larger choice; Silverman's game reverses that advantage once the larger-to-smaller ratio reaches the threshold.

Scope of Application

  • Game theory. Studies equilibrium and optimal mixed strategies.
  • Discrete variants. Use finite or countable positive-number sets.
  • Continuous variants. Use intervals or other uncountable sets.
  • Asymmetric variants. Give the two players different admissible sets.
  • Decision analysis. Illustrates a tradeoff between upward advantage and overshoot risk.

Clarity

State each player's strategy set, the threshold T, penalty nu, simultaneity assumption, ratio convention, complete payoff table, and the equilibrium concept used. Do not infer results across variants without checking these data.

Manages Complexity

The abstraction compresses a continuous or discrete choice space into three payoff regions: tie, advantageous moderate superiority, and penalized overshoot. This makes the nonmonotone strategic incentive explicit.

Abstract Reasoning

  1. Declare both strategy sets and parameters.
  2. Collect one independent choice per player.
  3. Order the choices and compute their ratio.
  4. Classify the ratio as tie, moderate, or overshoot.
  5. Assign the corresponding zero-sum payoff.
  6. Analyze strategies under the specified set and solution concept.

Knowledge Transfer

The transferable cargo is a thresholded comparison incentive: relative advantage becomes disadvantage beyond an overshoot boundary. It transfers literally only when positive choices, ratio regions, and zero-sum payoffs are preserved; otherwise it is an analogy.

Examples

Applied / In Practice

With T equal to 3 and choices 4 and 2, the larger-to-smaller ratio is 2, so the larger choice earns the winning payoff.

Mapped back: ratio → 2; region → between 1 and T.

Applied / In Practice

With T equal to 3 and choices 8 and 2, the ratio is 4, so the player choosing 8 receives the penalty rather than a win.

Mapped back: ratio → 4; region → at least T.

Applied / In Practice

Two players choose numbers and the larger always wins regardless of magnitude; the missing overshoot penalty makes this a different comparison game.

Mapped back: thresholded reversal → absent.

Structural Tensions

T1 — Larger Choice versus Overshoot Penalty. The game rewards being larger only within a bounded ratio.

Diagnostic: Where does the payoff reverse?

T2 — Strategy-Set Breadth versus Equilibrium Structure. Finite and continuous sets retain the rule but can support different solution behavior.

Diagnostic: Which admissible set is being analyzed?

T3 — Symmetry versus Asymmetric Extensions. Common strategy sets make players symmetric while variants may give them different sets.

Diagnostic: Does the variant preserve the same payoff comparison?

Structural–Framed Character

Silverman's Game is hybrid: structurally a threshold-partitioned zero-sum comparison and framed by game-theoretic strategy sets, payoffs, and equilibrium conventions.

Structural Core vs. Domain Accent

The core is simultaneous selection followed by a ratio test with a nonmonotone payoff. Game theory supplies strategy sets, mixed strategies, zero-sum evaluation, equilibrium questions, and variants across discrete and continuous carriers.

  • Approved root. The frozen review found no live node that necessarily contains this particular thresholded ratio game.

  • Related — zero-sum game, simultaneous game, mixed strategy, payoff matrix, and continuous game. These describe its game-theoretic setting without replacing its rule.

Neighborhood in Abstraction Space

Silverman's game sits in a crowded region of the domain-specific corpus (28th percentile for distinctiveness): several abstractions share nearly its structure, so a description that fits it tends to fit its neighbors too.

Family — Combinatorial Optimization & Game Problems (12 abstractions)

Nearest neighbors

Computed from structural-signature embeddings · 2026-10-08

Not to Be Confused With

  • Largest-Number Game. Tell: Always rewarding the larger choice omits the threshold penalty.
  • War of Attrition. Tell: Its costs depend on persistence or timing, not a ratio of selected positive numbers.
  • Auction. Tell: An auction allocates goods and payments rather than applying Silverman's payoff partition.
  • General Zero-Sum Game. Tell: This broader class does not require positive-number choices or an overshoot reversal.

References

  • Frozen Wikipedia discovery revision: https://en.wikipedia.org/wiki/Silverman%27s_game (revision 1021634893).
  • Preserved source candidate: http://math.ucsd.edu/~revans/Silverman.pdf
  • Preserved source candidate: https://www.springer.com/us/book/9783540592327

The frozen Wikipedia revision is discovery provenance. The retained source set was reviewed for identity, formal or operational relation, and scope. The encyclopedia's structural synthesis is bounded to those claims; a thin authority surface is recorded as a nonblocking source-strengthening repair rather than concealed.