The Application of Non-Cooperative Stackelberg Game Theory in Behavioral Science: Social Optimality with any Number of Players

Here we present a ground-breaking new postulate for game theory. The first part of this postulate contains the axiomatic observation that all games are created by a designer, whether they are: e.g., (dynamic/static) or (stationary/non-stationary) or (sequential/one-shot) non-cooperative games, and importantly, whether or not they are intended to represent a non-cooperative Stackelberg game, they can be mapped to a Stackelberg game. I.e., the game designer is the leader who is totally rational and honest, and the followers are mapped to the players of the designed game. If now the game designer, or “the leader” in the Stackelberg context, adopts a pure strategy, we postulate the following second part following from axiomatic observation of ultimate game leadership, where empirical insight leads to the second part of this postulate. Importantly, implementing a non-cooperative Stackelberg game, with a very honest and rational leader results in social optimality for all players (followers), assuming pure strategy across all followers and leader, and that the leader is totally rational, honest, and is able to achieve a minimum amount of competency in leading this game, with any finite number of iterations of leading this finite game. John Nash stated the following theoremfor non-cooperative games that became widely known as the Nash equilibrium:, Theorem 1 (Nash Equilibrium) Every game with a finite number of players and action profiles has at least one Nash equilibrium [Nash, 1951]. A mixed strategy always has an equilibrium. [Nash, 1951]. We have discovered a postulate as a counter-point to the Nash-Equilibrium Theorem 1 above. Our postulate is: ∗NICTA is funded by the Australian Government through the Department of Communications and the Australian Research Council through the ICT Centre of Excellence Program. Mixed implies non-deterministic i.e., a best response to all others’ best responses, which is not guaranteed for a completely deterministic, or pure, strategy.