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.
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.
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. Inclusion test: Require two players, declared positive-number strategy sets, independent choices, parameters T greater than one and nu greater than zero, and the tie/reward/overlarge-penalty payoff partition based on the ratio of their choices. Exclusion test: Exclude generic higher-number games, auctions, sequential guessing, payoff rules based on absolute difference, and games without the overlarge-choice penalty. Nearest boundary: 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. Exit condition: The identity fails when the thresholded ratio payoff is replaced, even if two players still choose numbers. Common misclassifications: 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. Nearest named distinctions: Largest-Number Game: Always rewarding the larger choice omits the threshold penalty. War of Attrition: Its costs depend on persistence or timing, not a ratio of selected positive numbers. Auction: An auction allocates goods and payments rather than applying Silverman's payoff partition. General Zero-Sum Game: This broader class does not require positive-number choices or an overshoot reversal.
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¶
- Declare both strategy sets and parameters.
- Collect one independent choice per player.
- Order the choices and compute their ratio.
- Classify the ratio as tie, moderate, or overshoot.
- Assign the corresponding zero-sum payoff.
- 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.
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
- Two-Moment Decision Model — 0.91
- Kingmaker Scenario — 0.89
- Graphical Game Theory — 0.89
- Probability matching — 0.88
- Dominant Factor Test — 0.88
Computed from structural-signature embeddings · 2026-10-08