The collection

Every essay — page 42

Page 42 of 42, continuing through the fields in the same order.

Geometry Analysis Algebra Discrete Topology Probability Number Dynamics Logic Computation Applied What's new Series Concepts Search

Applied

A rule for choosing, stated exactly, and what it forces on whoever adopts it.

Three random games and the cells where best replies meet. Three random 5 by 5 games with best replies marked; pure equilibria (cells best for both players): 0, 1 and 2.

A third of random games have no pure equilibrium

Where the equilibria of a random four-by-four game are found. 7 equilibria with supports (1, 1), (3, 2), (4, 3), (13, 12), (14, 13), (34, 23), (134, 123).

Every random game has an odd number of equilibria

Seven workers, seven jobs, one cheapest way. A 7 × 7 table of exponential costs; optimal assignment 1→5, 2→2, 3→4, 4→1, 5→6, 6→7, 7→3 costing 0.7482; expected optimum Σ 1/k² = 1.511797.

One, plus a quarter, plus a ninth

Fictitious play that never settles. Shapley's 3 × 3 game, 34 runs in 2,000,000 rounds; final mixtures row 0.371, 0.456, 0.173, column 0.253, 0.544, 0.203.

A best reply to the past

Where sellers stand when nobody wants to move. 2 sellers: 1 equilibria (e.g. 12,12 of 24); 3 sellers: 0 equilibria; 4 sellers: 3 equilibria (e.g. 6,6,17,18 of 24); 5 sellers: 4 equilibria (e.g. 4,4,12,19,20 of 24); 6 sellers: 25 equilibria (e.g. 3,3,9,14,20,21 of 24).

Sellers on a street stand back to back

6 figures · new

A party that loses every issue wins the vote. Groups A: 110, B: 101, C: 011, D: 000, E: 000; X wins 3–2; issues won by X: 0 of 3.

A party that loses every issue can win

5 figures · new