Navigating the Topology of 2x2 Games: An Introductory Note on Payoff Families, Normalization, and Natural Order

Understand

The Robinson-Goforth topology of swaps in adjoining payoffs elegantly arranges 2x2 ordinal games in accordance with important properties including symmetry, number of dominant strategies and Nash Equilibria, and alignment of interests.

• Adding payoff families based on Nash Equilibria illustrates an additional aspect of this order and aids visualization of the topology.
• Making ties through half-swaps not only creates simpler games within the topology, but, in reverse, breaking ties shows the evolution of preferences, yielding a natural ordering for the topology of 2x2 games with ties.
• An ordinal game not only represents an equivalence class of games with real values, but also a discrete equivalent of the normalized version of those games.