Rationalizability, Observability, and Common Knowledge

Abstract

We study the strategic impact of players’ higher-order uncertainty over the observability of actions in general two-player games. More specifically, we consider the space of all belief hierarchies generated by the uncertainty over whether the game will be played as a static game or with perfect information. Over this space, we characterize the correspondence of a solution concept which captures the behavioral implications of Rationality and Common Belief in Rationality (RCBR), where ‘rationality’ is understood as sequential whenever the game is dynamic. We show that such a correspondence is generically single-valued, and that its structure supports a robust refinement of rationalizability, which often has very sharp implications. For instance: (i) in a class of games which includes both zero-sum games with a pure equilibrium and coordination games with a unique efficient equilibrium, RCBR generically ensures efficient equilibrium outcomes (eductive coordination); (ii) in a class of games which also includes other well-known families of coordination games, RCBR generically selects components of the Stackelberg profiles (Stackelberg selection); (iii) if it is commonly known that player 2’s action is not observable (e.g., because 1 is commonly known to move earlier, etc.), in a class of games which includes all of the above RCBR generically selects the equilibrium of the static game most favorable to player 1 (pervasiveness of first-mover advantage).

The value of a coordination game

“…20 On the other hand, mixed Nash equilibrium predicts miscoordination but cannot account for coordination failure; moreover, it has unattrac-20 Other prominent evolutionary models select the efficient equilibrium (Robson and Vega-Redondo, 1995) or select different equilibria depending on the economic environment (Binmore and Samuelson, 1997), features of the learning process (Crawford, 1995), or on initial conditions (Samuelson, 2002). There are also equilibrium refinements that select the efficient outcome, such as payoff dominance (Harsanyi and Selten, 1988) or refinements that require predictions to be robust to perturbing the assumption that the extensive form is common knowledge (Penta and Zuazo-Garin, 2022).…”

Information design in multistage games

“…The second insight is that our characterization of Bayes' correlated equilibrium in multistage games generalizes to any solution concept for which a revelation principle 1 Our work is related in spirit to Penta (2015) and Penta and Zuazo-Garin (2021). The first paper extends the belief-free approach to robust mechanism design to dynamic problems, in which agents obtain information over time about payoff-relevant states, but where actions are public.…”

“…The second paper considers the question of robust predictions when common knowledge assumptions about players' information on each other moves are relaxed; payoffs remain common knowledge. In contrast to our paper, where we consider all expansions, Penta and Zuazo‐Garin (2021) incorporate only “local” perturbations of the belief hierarchies.…”

“…11 Our work is related in spirit to Penta (2015) and Penta and Zuazo‐Garin (2021). The first paper extends the belief‐free approach to robust mechanism design to dynamic problems, in which agents obtain information over time about payoff‐relevant states, but where actions are public.…”

Behavioral equivalence of extensive game structures

“…For example, only the first mover can be afraid (or hopeful) of being spied on. More generally, we can cleanly represent incomplete information about the order of moves and the information structure as in [21], whereas the representation à la Kuhn [15] would be cumbersome.…”