♟️ Nash Equilibrium — The Maths of Strategy

For each column the code compares the row player's two payoffs in that column and marks the larger one (rowBRset); for each row it compares the column player's two payoffs in that row (colBRset). If both numbers in a comparison are equal it treats both strategies as tied best responses. A cell is boxed as a pure Nash equilibrium only when the row payoff is a best response down its column AND the column payoff is a best response across its row at the same time.

The simulation solves the indifference conditions directly from the payoff numbers. It finds p (the probability the row player plays their first strategy) so that the column player is indifferent between their two options, and q (the probability the column player plays their first strategy) so that the row player is indifferent. Algebraically p = (d22-d21)/(d11-d12-d21+d22) using the column player's payoffs, and q is the mirror formula using the row player's payoffs. If either denominator is zero, or the result falls outside [0,1], no valid mixed equilibrium exists and the stat shows a dash.

Best-response dynamics in this simulation is a fixed alternating process: on every step the row player (or column player, alternating by turn) jumps to their current best response, regardless of whether the resulting cell is an equilibrium. Once the token lands on a boxed cell, both players' best responses point back at that same cell, so the token simply keeps landing there step after step — it looks stationary because it genuinely is, but the loop never explicitly "stops."

In Matching Pennies every cell has exactly one player wanting to switch, so there is no pure Nash equilibrium at all in this preset. Best-response dynamics therefore cycles through all four cells (Heads,Heads) → (Heads,Tails) → (Tails,Tails) → (Tails,Heads) → back to start, forever. The only equilibrium is the mixed one at p = q = 0.5, visible as the yellow dot at the centre of the correspondence view, which the alternating pure-strategy dynamics can never actually land on.

It draws p (probability row plays their first strategy) against q (probability column plays their first strategy) on a unit square. The green curve is the row player's best response p*(q) for every possible q, and the blue curve is the column player's best response q*(p) for every possible p, each computed directly from the expected-payoff formulas in the code. Blue dots mark the pure equilibria at the square's corners; a yellow dot marks the mixed equilibrium wherever the two curves genuinely intersect inside the square.