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The importance of higher-order beliefs to successful coordination

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


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.

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  1. See also a similar finding by Baeriswyl and Cornand (2014).

  2. Shapiro et al. (2014) and Kneeland (2016) explore limited depth of reasoning in coordination games.

  3. Indeed Anctil et al. (2004) find that subjects play the risk-dominant equilibrium in a coordination game with payoff uncertainty, but also find similar play of this equilibrium in a control treatment without payoff uncertainty.

  4. More recent work on stag hunt games indicates that subjects might invest even when they expect coordination to fail in order to shift future play towards investment in later rounds of a fixed matching (Hyndman et al. 2009). Cheap talk conveying social disapproval can also induce subjects to coordinate better (Dugar 2010). Coordination is generally more likely when equilibrium payoffs are more equal (López-Pérez et al. 2015). Polonio et al. (2015) show that the way that subjects process visual information leads them to select different strategies in stag hunt games.

  5. I.e. repeated matchings were possible. This was communicated to subjects.

  6. This design was chosen primarily to minimize the concern that subjects can hedge by stating beliefs different from their true ones. Refer to Blanco et al. (2010) for a discussion. In the data beliefs are rather positively correlated with investment choices, suggesting that subjects did not perceive significant hedging opportunities.

  7. Though it is conceivable that the order of belief elicitation and action choice matters, the sequence of choice and then belief elicitation was identical across treatments. This was done to simplify the exposition of the instructions. Furthermore, had beliefs and choices been entered on the same computer screen, limitations arising from the nature of human–computer interaction ensure that one of these would have been entered before the other.

  8. Care was taken to implement the “decontextualized” context expected of economic experiments. The terms “optimistic” and “pessimistic” were not used in the instructions, nor was “invest”. Rather subjects were asked to indicate “how many of the [e.g.] 9 people who saw their partner’s coin to be Heads chose A”.

  9. We did not elicit counterfactual beliefs about how many people would have invested if they had known their partner's investment frequency as no one in Treatments IK and NK ever had this information—making incentivized elicitation impossible.

  10. Since all subjects are matched randomly we can think of this number as the probability that one’s partner will choose to invest since partners are a random draw from this set of 19. This method of belief elicitation is preferable to a proper scoring rule (e.g. quadratic scoring) because scoring rules are only incentive compatible under very specific forms of risk preferences. Incentivizing only correct guesses is robust to risk aversion. A proof of the incentive-compatibility of this elicitation mechanism may be found in Appendix 1.


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The author wishes to thank professor Lise Vesterlund for advice and support, professor Stephanie Wang for help in experimental design, seminar participants at the University of Pittsburgh, attendees at the 2012 Economic Science Association World and North American meetings, and participants at the 2015 Thurgau Experimental Economics Meeting. Support was also received from the Kiel Institute for the World Economy. Anonymous referees made several helpful suggestions towards improving the exposition of the paper.

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Correspondence to Steven J. Bosworth.

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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

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Bosworth, S.J. The importance of higher-order beliefs to successful coordination. Exp Econ 20, 237–258 (2017).

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