Silviase/jgov_v1.4_full_ann huggingface dataset:
fundamentals brings about a unique equilibrium. Their unique result is surprising, and clarifies the response of speculators to the signal that completely changes from an attacking to a not-attacking position at a critical point in the states of fundamentals.

We first show that multiple equilibria can never be eliminated in the game considered here, in which each individual considers the choices of other individuals, even though a small amount of noise is perturbed to the signal about fundamentals, as in Morris and Shin (1998, 2001). Moreover, not only the unproductive—that is, non-producing equilibrium—but also an unnatural equilibrium—such that individuals choose to produce when signals are in some interim interval but not to produce when it is in the upper region—is also permissible.

Despite the negative counterintuitive result, our first result ironically gives an important implication. The above result is due to the lack of a dominant action in the highest region of fundamentals. That is, a decision to produce is not necessarily dominant in the highest region by the structure of the model. On the other hand, in Morris and Shin (1998), it is stipulated as common knowledge that, irrespective of the choice of others, the action to attack and the action not to attack are dominant when the state of fundamentals is in the lowest and highest regions respectively. This assures the uniqueness of equilibrium, and shows that the uniqueness of equilibrium is realized by the common belief that other individuals surely choose to attack and not to attack at the extreme values of fundamentals. The mechanism is also common to the results of Frankel (2003).

Therefore, to guarantee a productive equilibrium as a unique equilibrium in our model, production must somehow be dominant in the highest region of fundamentals. Conversely, once the dominance of production is locally secured in the region, a productive equilibrium, where all individuals in a large region of fundamentals do produce, is guaranteed. That is, a productive equilibrium is comprehensively obtained only by the local dominance of production in the extreme region

This local dominance of production is specifically represented by a positive
payoff to the production of individuals, who observe the signal in the highest states of fundamentals in our model. Then it can be obtained if it is commonly believed that the majority of individuals who observe the signal equal to unity produce. Further, we show it can be assured also by a production subsidy given only to the highest small region in the states of fundamentals. Government interventions then restore productive equilibrium.

Next, we extend our basic framework to the model that contains individuals who make unsophisticated choices as well as those who make rational choices. In our basic model, all individuals are supposed to behave in a Bayesian manner, i.e. an individual rationally chooses whether to produce based on the computation of the expected payoff to production conditional on the noisy signal about fundamentals, taking the strategic interactions between players into consideration. It appears that in the real world, however, some people behave under the strong influence of the prevailing belief or sensation for economic conditions. Moreover, other people make decisions based on their false beliefs and misperceptions, as the noise traders modeled by De Long et al. (1990) $ ^{4} $ .

In the extended model, we consider an economy that consists of those unsophisticated individuals as well as rational ones. Then we show that in the case in which unsophisticated individuals make decisions based simply on the prevailing atmosphere for fundamentals, only the productive equilibrium appears under optimistic conditions, but multiple equilibria occur under pessimistic conditions. Further, in the case in which unsophisticated individuals are those who falsely and always take the signals at their face values and make decisions based only on what the signals just told them, the appearance of productive equilibrium is guaranteed independently of the conditions.

In our model a positive payoff to production in the highest states of
fundamentals, is never guaranteed a priori. In the extended model with unsophisticated players, however, unsophisticated players facilitate a positive payoff to the production of rational individuals, who observe the signal in the highest states, and only productive equilibrium occurs. It is interesting to note that the existence of unsophisticated players may well serve rather than disrupt the stability of an economic system with coordination problems. In particular, it is noteworthy in our case that the naive contribution of unsophisticated players only in the restricted, local, region of fundamentals is sufficient for productive equilibrium of production in a large region of fundamentals.

The rest of this paper is organized as follows. Section 2 presents the basic model and briefly reviews the multiple equilibria in perfect information. Section 3 shows the result of the basic model in imperfect information, in which the prior distribution of the state of fundamentals and the distribution of noise are uniform. Section 4 presents the result of the extended model to the case with noisy or unsophisticated participants, in which the prior distribution of the state and the distribution of noise are generalized. Section 5 concludes the argument and the Appendix collects all proofs.

## 2 Game with Perfect Information in Basic Model

### 2.1 Basic Model

The essence of the coordination problem addressed in Diamond (1982) is simply modeled by the following game in a single population (for a setting in two populations, see section 4.7 of Vega-Redondo (1996)). We consider an economy with a continuum of anonymous individuals, of unit mass. Each individual chooses whether to produce y units of consumption good with production cost c.

When an individual produces y units of consumption good, he cannot by himself consume the goods that he produces. Moreover, for deriving utility from them, he has to trade them for the goods another individual produces.
Therefore, an individual who has produced something is given an opportunity to trade with another individual by random matching. Then, if the individual he meets has also produced, he trades the goods with the partner and obtains payoff y-c. But if the partner has not produced, his payoff is -c. If an individual chooses not to produce, his payoff is fixed to be zero at the time.

Each individual is assumed to be risk neutral and to know the structure of these payoffs. Moreover, we assume that when producing is indifferent to no producing, an individual chooses not to produce. Denoting by strategy 1 and strategy 2—choosing to produce and not to produce—the above setting is represented as the population game based on the following symmetric payoff matrix: $ \begin{pmatrix}(y-c,y-c)&(-c,0)\ $ 0,-c)&(0,0)\end{pmatrix} $ .

We now introduce a state of fundamentals into the model. The economy is characterized by a state of fundamentals $ \theta\in[0,1] $ , and the number of units of the produced good y and the production cost c are functions of $ \theta $ . As $ \theta $ goes up, the fundamentals get better, and potential productivity as well as potential demand of the economy are larger. We then assume that $ y(\theta) $ and $ c(\theta) $ are continuously differentiable and satisfy the following conditions.

A1. $ c(0) > y(0) > 0 $

$$ A2.\ y(1)>c(1)>0 $$

$$ \mathrm{A}3.\ y^{\prime}(\theta)>c^{\prime}(\theta)>0,\ \forall\theta\in[0,1]. $$

These conditions ensure that $ y(\theta) $ and $ c(\theta) $ are strictly increasing in $ \theta $ and there exists a unique $ \theta \in (0,1) $ such $ y(\underline{\theta}) = c(\underline{\theta}) $ . Our model differs from those of Morris and Shin (1998, 2001) in that even in the highest states of fundamentals, a positive payoff to production is never assured if the proportion of individuals to produce is sufficiently small. As will be seen below, this brings about a sharp difference for the uniqueness of a solution in a game with imperfect information.

### 2.2 Game with Perfect Information
First, we consider the game with perfect information of fundamentals in which each individual knows exactly the true value of $ \theta $ . When the state of fundamentals is $ \theta $ and the proportion of individuals who produce is $ l \in [0,1] $ , the expected payoff to the production of an individual is $ \pi(l,\theta) = ly(\theta) - c(\theta) $ .

For a given $ \theta $ , denote by $ l^{*}(\theta) $ the proportion of individuals who produce in equilibrium. Then it holds that $ l^{*}(\theta)=1 $ if $ \pi(l^{*},\theta)>0 $ and $ l^{*}(\theta)=0 $ otherwise. Further, by the assumption, $ y(\theta)-c(\theta)>0 $ if and only if $ \theta>\theta $ . Hence we have the following for the equilibria of the game:

$$ l^{*}(\boldsymbol{\theta})=\begin{array}{c}0\quad if\ 0\leq\boldsymbol{\theta}\leq\underline{\boldsymbol{\theta}}_{}\\ 0,1\quad if\ \underline{\boldsymbol{\theta}}<\boldsymbol{\theta}\leq1\end{array}. $$

Under the assumption of perfect information of fundamentals, all individuals always know the exact value of $ \theta $ . Therefore, when $ 0 \leq \theta \leq \theta $ , each individual knows that $ \theta $ is in the region where the expected payoff to production is non-positive, even if all individuals produce. Then all individuals refrain from producing in the equilibrium.

On the other hand, when $ \theta < \theta \leq 1 $ , each individual knows that the expected payoff to production is positive if the proportion of individuals who produce is beyond $ \frac{c(\theta)}{y(\theta)} $ , and non-positive otherwise. In this context the choice of an

individual depends on his belief about the choice of other individuals. That is, if an individual believes that sufficiently many individuals choose to produce, a decision to produce is rational and otherwise a decision not to produce is rational. Thus, there occur multiple equilibria in which all individuals produce or all individuals do not produce in the region.

## 3 Game with Imperfect Information in Basic Model

In this section we consider a game with imperfect information in which each individual observes a noisy signal about a state $ \theta $ and, based on it, chooses whether or not to produce.
### 3.1 Structure and Assumption

We suppose that the state $ \theta $ is uniformly distributed on [0,1] and that when the state is $ \theta $ , each individual is given a private signal $ x = S(\theta + \sigma\varepsilon) $ ,

$$ 0if 冮 x<0 $$

where $ \sigma > 0 $ and $ S(x) = x if 0 \leq x \leq 1 $ . The noise term $ \varepsilon $ is also uniformly

1 if 1 < x

distributed on $ [-1,1] $ and independent of $ \theta $ . Further, it is independent across individuals. Here we technically remark that x=0,1 have a point mass and need to be treated differently from other points. We denote the cumulative distribution function of $ \theta $ by F.

Each individual exactly knows about the distribution of $ \theta $ and x, and the game proceeds according to the following steps:

1. Nature selects a state $ \theta $ .

2. Each individual observes a private signal x.

3. Based on the signal x, each individual decides whether to produce.

4. Each individual carries out his decision and obtains the payoff.

We assume production function $ y(\theta) $ and cost function $ c(\theta) $ are continuously differentiable and satisfy the following conditions:

A1. $ c(0)>y(0)>0 $ .

A2'. $ y(1) > 4c(1) > 0 $

A4'. $ y(\theta) $ is concave and $ c(\theta) $ is convex.

These conditions are met, for example, by $ y(\theta)=4\theta, c(\theta)=\frac{1}{4}(2\theta+1) $ . In the above, A2' and A3' are to guarantee a higher payoff in the upper region than A2 and A3 in section 2.1.

For a given profile of strategies of individuals, the aggregated strategy of individuals is represented by a $ [0,1] $ -valued function s on $ [0,1] $ , where $ s(x) $ is the proportion of individuals who produce when the value of the signal is x.
Hereafter, we always assume the measurability of s.

As signals are uniformly distributed on $ [\theta-\sigma,\theta+\sigma] $ when the value of the state is $ \theta\in[\sigma,1-\sigma] $ , we have $ l(\theta,s)=\frac{1}{2\sigma}\int_{\theta-\sigma}^{\theta+\sigma}s(x)dx $ . Then, for a given strategy s, the payoff to an individual who produce at state $ \theta $ is $ \pi(l(\theta,s),\theta)=\frac{y(\theta)}{2\sigma}\int_{\theta-\sigma}^{\theta+\sigma}s(x)dx-c(\theta) $ for $ \theta\in[\sigma,1-\sigma] $ . Further, since the posterior distribution of $ \theta $ when an individual observes signal $ x\in[\sigma,1-\sigma] $ is the uniform distribution on $ [x-\sigma,x+\sigma] $ , the expected payoff to production conditional on signal x is given by

$$ u(x,s)=\frac{1}{2\sigma}\int_{x-\sigma}^{x+\sigma}\pi\left(l(\theta,s),\theta\right)d\theta\\ =\int_{x-\sigma}^{x+\sigma}\frac{d\theta}{2\sigma}[\frac{y(\theta)}{2\sigma}\int_{\theta-\sigma}^{\theta+\sigma}s(x)dx-c(\theta)] $$

for $ x \in [\sigma, 1 - \sigma] $ . This formula is slightly modified for $ x \in (1 - \sigma, 1) $ as

$$ u(x,s)=\int_{x-\sigma}^{1}\frac{d\theta}{1-x+\sigma}[\frac{y(\theta)}{2\sigma}\int_{\theta-\sigma}^{\theta+\sigma}s(x)d x-c(\theta)], $$

and similarly for $ x \in (0, \sigma) $ .

In the case of perfect information, the choice of each individual at state $ \theta $ depends only on that of other individuals at $ \theta $ , and it is independent of those at other states. On the other hand, in the case of imperfect information, the choice of each individual who observes signal x depends on that of other individuals whose signals are in $ [x-2\sigma, x+2\sigma] $ through the above computation of $ u(x, s) $ .

We now state some definitions about the equilibria of the game.

Definition 1: The (aggregated) strategy s is optimal in the game if and only if $ s(x)=1 $ whenever $ u(x,s)>0 $ and $ s(x)=0 $ whenever $ u(x,s)\leq0 $ .

Definition 2: An equilibrium of the game is a profile of strategies of individuals that represents an optimal aggregate strategy.
When s is optimal, s takes values in $ \{0,1\} $ . Clearly $ s \equiv 0 $ is always an optimal strategy, and the corresponding equilibrium is called unproductive:

Definition 3: An unproductive equilibrium is a profile in which each individual always chooses not to produce.

In contrast, an equilibrium that represents an optimal strategy, of which form is of $ 1_{(k,1]} $ for some $ k \in (0,1) $ , is called productive.

Definition 4: A productive equilibrium is a profile that represents an optimal strategy in which each individual chooses to produce if and only if his private signal is above some $ k \in (0,1) $ .

As will be seen later, in general not only the unproductive equilibrium but an unnatural equilibrium such that $ s(x_{1})=1 $ and $ s(x_{2})=0 $ for some $ (x_{1},x_{2}) $ with $ x_{1} 0 $ in a self-fulfilling way if it is commonly believed that $ s(1) > \frac{3c(1)}{y(1)} $ . Because $ u(1, s) > 0 $ , which means $ s(1) = 1 $ , always holds for any sufficiently small $ \sigma > 0 $ if all individuals commonly believe that strictly more than $ \frac{3c(1)}{y(1)} $ of individuals whose signal is one always choose to produce. In the below, by “under condition (RC),” we mean “among all strategies that satisfy $ s(1) = 1 $ .”

### 3.2 Results

For the statement of the following proposition, we denote by $ l(\theta, \sigma) $ the proportion of individuals who end up producing at the state $ \theta $ . Further, we
define social welfare $ W(\theta, \sigma) $ and its total value $ \overline{W}(\sigma) $ by $ W(\theta, \sigma) = l(\theta, \sigma)(y(\theta)l(\theta, \sigma) - c(\theta)) $ and $ \overline{W}(\sigma) = \int_{0}^{l} W(\theta, \sigma)d\theta $ respectively.

Proposition 1: Under condition (RC), for any $ \sigma\in(0,\sigma_{0}) $ where $ \sigma_{0}=[\sup\{\sigma>0\}|y(1-2\sigma)>c(1)\}]\wedge\frac{\theta}{2} $ , there exists a unique equilibrium such that all individuals whose signal $ x\leq k(\sigma) $ decide not to produce, and all individuals whose signal $ x>k(\sigma) $ decide to produce for some $ k(\sigma)\in(0,1) $ .

Consequently, the following hold for l and W in the outcome, and we have

$$ \overline{{W}}=\int_{k-\sigma}^{k+\sigma}\frac{1}{2\sigma}(\theta+\sigma-k)(\frac{y(\theta)}{2\sigma}(\theta+\sigma-k)-c(\theta))d\theta+\int_{k+\sigma}^{1}\left(y(\theta)-c(\theta)\right)d\theta. $$

$$ \begin{array}{r}{(i)\quad l=0\mathrm{a n d}\quad W=0\quad\mathrm{i f}\quad0\leq\theta\leq k-\sigma~.}\end{array} $$

$$ \left(\mathrm{i i}\right)l=\frac{1}{2\sigma}(\theta+\sigma-k)\mathrm{a n d}W=l(\theta,\sigma)(y(\theta)l(\theta,\sigma)-c(\theta))\mathrm{i f}k-\sigma<\theta0 $ if $ \theta^{*}<\theta0\quad\mathrm{i f}\quad k+\sigma\leq\theta\leq1. $$

Explanation: As already mentioned, in general we can not eliminate unproductive equilibrium. However, the first part of Proposition 1 implies that the regular condition assures that the productive equilibrium is a unique solution. That is, under the situation that all individuals whose signal is unity always choose to produce and it is commonly believed, the productive equilibrium is a unique equilibrium. This result is explained as follows.

Suppose that an individual considers whether to produce when he observes signal $ x_{1} \in (\underline{\theta}, 1) $ . In this case he calculates the conditional distribution of signals conditioned on his private signal $ x_{1} $ , and then estimates the expected
payoff to production with the responses of other individuals to signal $ x \in [x_{1} - 2\sigma, x_{1} + 2\sigma] $ . So, he has to expect the responses of other individuals to signal $ x \in [x_{1} - 2\sigma, x_{1} + 2\sigma] $ for that. But they depend on the responses to signal $ x \in [x_{1} - 4\sigma, x_{1} + 4\sigma] $ , and so on. Thus, he reaches to expect the responses of other individuals to signal $ x \in [\underline{\theta} - 2\sigma, 1] $ , since it is commonly known that individuals whose signal is below $ \underline{\theta} - 2\sigma $ surely choose not to produce.

At this stage, an optimal strategy for individuals is uniquely obtained by a “backward induction” when the boundary condition $ s(1)=1 $ is given. To begin with, positive $ u(x,s) $ in a neighborhood of 1, say $ (1-\varepsilon_{1},1] $ , is assured by $ s(1)=1 $ . In turn, this ensures positive $ u(x,s) $ in $ (1-\varepsilon_{1}-\varepsilon_{2},1] $ for some small $ \varepsilon_{2}>0 $ . By continuing this procedure, the turning point $ k(\sigma) $ , where individuals whose signal $ x>k(\sigma) $ choose to produce, and individuals whose signal $ x\leq k(\sigma) $ choose not to produce, is uniquely determined and $ s=1_{(k(\sigma),1]} $ is obtained as a unique optimal strategy for individuals, since y-c is continuous, strictly increasing and non-positive on $ [0,\theta-2\sigma] $ . This is why only the productive equilibrium, where all individuals in the large region of fundamentals do produce, occurs as a snow slide takes place, once it is commonly believed that all individuals whose signal is unity always choose to produce.

(ii) In the second part of proposition 1 ensures that when $ \theta $ , the state of fundamentals, is within $ (k(\sigma)-\sigma,\theta^{*}) $ , a proportion, $ \frac{1}{2\sigma}(\theta+\sigma-k(\sigma)) $ , of individuals will end up producing and suffer losses. This superficially seems paradoxical to the notion that each individual optimizes his choice to the private signal. Why does this occur?

As later seen in the proof, when $ k(\sigma) - \sigma < \theta < \theta^{*}, l(\theta, I_{(k(\sigma),1]}) < \frac{c(\theta)}{y(\theta)} $ and then $ \pi(l,\theta) < 0 $ . That is, the proportion $ l(\theta, I_{(k(\sigma),1]}) $ of individuals who observe the signal x beyond $ k(\sigma) $ and choose to produce is so small that the expected payoff to production is negative. But their choices to produce
are rational in the Bayesian sense, because the expected payoff to the production of the individual is positive, which easily follows from that $ u(x,I_{k(\sigma)}) $ is strictly increasing for $ k(\sigma)-\sigma \leq x \leq k(\sigma)+\sigma $ and $ u(k,I_{k})=0 $ .

The limit value of $ k(\sigma) $ when $ \sigma $ goes to zero is easily obtained as a corollary by taking the limit as $ \sigma \to +0 $ in

$$ u(k,I_{(k,1]})=\int_{-1}^{1}\ \frac{dz}{2}[\frac{1-z}{2}\ y(k-\sigma z)-c(k-\sigma z)]=0\;. $$

Thus we can state the next corollary:

Corollary 1: $ \lim_{\sigma\to0}k(\sigma) $ is the solution of $ \frac{1}{2}y(x)-c(x)=0 $ .

Remarks: Corollary 1 implies $ k(\sigma) > \theta $ for any sufficiently small $ \sigma > 0 $ . This shows that even the productive equilibrium is inefficient in imperfect information. Because with imperfect information each individual whose signal x is in $ (\theta, k) $ never produces, although with perfect information the state that all individuals produce is possible when state $ \theta $ is in $ (\theta, k) $ .

In intuition, Corollary 1 is due to the fact that when the value of his private signal is k and all individuals behave based on $ I_{(k,1]} $ , he expects that half of individuals choose to produce, since the prior distribution is uniform, as pointed out in Morris and Shin (2001).

If condition (RC), which substitutes for the assumption of the upper dominance region in Morris and Shin (1998, 2001), always holds, only the productive equilibrium appears by proposition 1. This implies that there is room for a policy treatment to guarantee good equilibrium by ensuring condition (RC). Proposition 2, which immediately follows as a corollary of proposition 1 from
$$ u(1,s)=\int_{1-\sigma}^{1}\ \frac{2(\theta+\sigma-1)}{\sigma^{2}}d\theta[\frac{1}{2\sigma}(\int_{\theta-\sigma}^{1}\ s(x)dx+(\theta+\sigma-1)s(1))y(\theta)-c(\theta)], $$

indicates how to realize it by a policy measure.

Proposition 2: Suppose it is commonly known that when the state $ \theta $ is in $ (1-\sigma,1] $ , any individual who produces is given the subsidy of $ (c(\theta)+\lambda) $ for any $ \lambda>0 $ . Then $ s=I_{(k(\sigma),1]} $ is a unique optimal strategy for all the individuals.

Explanation: Proposition 2 tells us that it suffices that a subsidy is given only when state $ \theta $ is in the highest region $ (1-\sigma,1] $ , where $ \sigma $ is the width of the error term. Also, $ \lambda $ can be a very small number. That is, the announcement of making up the production cost in the highest small region ensures the condition (RC) that individuals who observe the signal equal to unity surely choose to produce. Then, this guarantees the appearance of only the productive equilibrium, where all individuals in a large region of fundamentals do produce, by backward induction through individuals as already stated. The important point is that the announcement of a policy treatment is required only for the restricted region of fundamentals to obtain the productive equilibrium comprehensively.

This proposition implies that as long as the government provides the small group of individuals who obtain signals near the top with a little more than the production cost, then productive equilibrium will emerge. This can be done only if the government can verify the signal and only if the government can collect tax revenues by lump-sum taxes, of which the regressive property might be discussed in terms of the fairness of taxation.

If the signal is not verifiable even ex post, then the government should engage in production to ensure the condition (RC) if it can know the true value of $ \theta $ , or the government should hire some individuals who will produce when the signal is unity. Here the publicly hired individuals must be honest and must to be believed to be honest. Here, we see a resurrection of J. M. Keynes' idea of "pump-priming expenditure theory," because the required amount of subsidies for production can be rather small. As will be indicated