The importance of higher-order beliefs to successful coordination

“He who does not trust enough, Will not be trusted” – Lao Zi

Abstract

Beliefs about other players’ strategies are crucial in determining outcomes for coordination games. If players are to coordinate on an efficient equilibrium, they must believe that others will coordinate with them. In many settings there is uncertainty about beliefs as well as strategies. Do people consider these “higher-order” beliefs (beliefs about beliefs) when making coordination decisions? I design a modified stag hunt experiment that allows me to identify how these higher-order beliefs and uncertainty about higher-order beliefs matter for coordination. Players prefer to invest especially when they believe that others are “optimistic” that they will invest; but knowledge that others think them unlikely to invest does not cause players to behave differently than when they do not know what their partners think about them. Thus resolving uncertainty about beliefs can result in marked efficiency gains.

Appendices

Appendix 1: Incentivizing correct belief revelation

Suppose that a subject holds the belief that, on average, there is probability p that others will choose to invest. Denote the realized number of n other people who will choose to invest by m _a and the guess submitted by the subject m _g . We can then write the subject’s payoff from Estimation Task 1 as

$$ \pi_{ 1} = P(m_{g} = m_{a} ) \cdot u\,\left( {\$3 {\text{ or }}\$1. 50} \right) $$

where u is the subject’s utility of money. Since partners are ex-ante identically drawn from the population of other people in the room, the probability above is binomial:

$$ E\left( {\pi_{ 1} } \right) = P_{\text{BIN}} (n,p; \, m_{g} ) \cdot u\left( \cdot \right). $$

Maximizing expected payoff with respect to m _g yields the mode of the binomial distribution:

$$ m_{g} = {\mathbf{ \lfloor }}({\text{n}} + 1)p{\mathbf{ \rfloor }} $$

on all but an unmeasurable set of possible p. Since the experimenter observes n, it is possible to identify p as lying within a reasonably small interval.

1.1 Second-order elicitation

A similar argument will be applied to the second-order belief elicitation, but we will need to maintain a more restrictive expected utility assumption. As actions could be considered binomial, we may consider responses to Estimation Task 1 to be multinomial with n trials (other people) and n + 1 possible responses (m _g ∈ {0, 1, … , n}). Denote the probability of each response p ₀, p ₁, … , p _n with ∑p _i  = 1. Subjects must guess the number of people giving each response, denote this vector g = g ₀, g ₁, … , g _n with ∑g _i  = n. Denote the vector of actual responses to Estimation Task 1 a = a ₀, a ₁, … , a _n with ∑a _i  = n. The subject’s expected payoff from Estimation Task 2 is

$$ E\left( {\pi_{ 2} } \right) = \sum P(g_{i} = a_{i} ) \cdot u\left( {\$ 0. 2 5} \right). $$

If we maximize with respect to g we simply get the modes of the marginal distributions (as the utility is linear in probabilities):

$$ g_{i} = {\mathbf{ \lfloor }}(n + 1)p_{i} {\mathbf{ \rfloor }}. $$

Appendix 2: Belief updating

Table 8 displays estimated parameters from equations explaining elicited first-order beliefs. In all of these equations, current-period beliefs are regressed on last period’s beliefs and whether or not one’s partner invested, interacted with the information conditions prevailing in that round. Round fixed-effects are estimated. Since this is a lagged dependent variable model, the equation is estimated by two-stage least squares (Cameron and Trivedi 2005). Round interactions are added since prior beliefs become more important relative to new information in later rounds. Since beliefs about both optimistic and pessimistic subjects are elicited each round, these appear as separate equations for treatments BK and PK. Results are as we would expect given reasonable belief updating. Subjects whose partners invest are more likely to think that other people invest, though they discount observations of optimistic or second-order-optimistic partners investing.

Table 8 Instrumental variables regressions of beliefs on observable factors