As an academic, at least for me, getting a paper rejected at a journal is nothing new. Whenever a paper is rejected, the same fundamental question needs to answered: rework the paper and try again at different journal, or to abandon it altogether, pivot, and work on some new project. Doubtless, this fundamental question—push through or pivot—makes its appearance in a variety of settings, including people stuck in dead-end jobs wondering if they should pivot to a new field, or entrepreneurs wondering whether to push through or pivot after receiving highly ambiguous early-stage performance feedback. While the question was motivated by a personal desire to understand my own choices (of whether to abandon a project or not), it is perhaps most economically significant for entrepreneur.

In the context of entrepreneurship, decades of behavioral and economic literature document that entrepreneurs rarely evaluate signals with cold objectivity; instead, they routinely exhibit a profound degree of optimism and “irrational exuberance.” Standard economic doctrine views this cognitive distortion as a systemic market failure—a psychological defect that anchors founders to failing propositions, drives sunk-cost fallacies, and ultimately precipitates costly business liquidations.

Below we offer a model of push-vs-pivot decisions along with entrepreneur optimism/pessimism, and we analyze the economic consequences of miscalibrated beliefs (with the hope that the lessons here would even be of value to an academic deciding on the future course of a project!)

Spoiler alert:

So maybe, I should exhibit some degree of irrational exuberance and continue working on a project?

Model

Consider an entrepreneur who is working on an idea. We shall consider a two period setting. In the first period, the entrepreneur exerts effort without knowing the productivity \(\theta\) of the effort. Specifically, she knows that the value of the idea is \(\theta+\epsilon_{1}\), where \(\theta\) is the (intrinsic) productivity of the idea and \(\epsilon_{1}\) is random noise. We shall assume that ex-ante \(\theta\) is unknown, and in specific, that the entrepreneur believes that it is distributed normally with mean \(\mu_{e}\) and standard deviation \(\sigma\).

Past literature suggests that entrepreneurs often suffer from optimism. And in specific, that their ideas are possibly better on average compared to other similarly situated people. Let us capture this parsimoniously, by assuming that the true distribution of \(\theta\) is normal with mean \(\mu\) and standard deviation \(\sigma\). Thus, \(\mu_{e}>\mu\) (\(\mu_{e}<\mu\)) represents an optimistic (pessimistic) entrepreneur.

The timeline of our model is as follows: At the end of first period, she sees the value so far \(K=\theta+\epsilon_{1}\), and has to choose to push through and continue with the same approach, to obtain the final value \(2\theta+\epsilon_{2}\), or to pivot to a brand new idea (with a new \(\hat{\theta}\)) and obtain \(\hat{\theta}+\epsilon_{3}\). Note that we assume the \(2\theta\) term to represent the fact that the effort is cumulative. Finally, to fix ideas, assume that all the random noises \(\epsilon_{\cdot}\) are normally distributed with mean \(0\) and standard deviation \(\tau\).

The answer to whether they should push through or pivot depends on the relative size of \(E\left[2\theta+\epsilon_{2}|\theta+\epsilon_{1}=K\right]\) and \(E\left[\theta+\epsilon_{3}\right]\). That is, on \(2E\left[\theta|\theta+\epsilon=K\right]\) versus \(E\left[\theta\right]\). Given the normal-normal model, it is easy to show that the entrepreneur should push through and continue iff \(2\left(\mu_{e}+\frac{\sigma^{2}}{\sigma^{2}+\tau^{2}}\left(K-\mu_{e}\right)\right)\ge\mu_{e}\). Specifically, the optimal decision rule becomes

The decision rule reveals two interesting patterns

Optimism flight (when \(\sigma>\tau\)): Because they are optimistic about all of their ideas, their subjective valuation of the next unknown venture (\(\hat{\theta}\)) skyrockets. A mediocre first-period signal on the current project isn’t enough anymore. They think, “My next idea is going to be a blockbuster, so why waste time grinding out this okay one?” They pivot because the grass looks incredibly green on the other side.

Optimism trap (\(\tau>\sigma\)): When a bad signal arrives, the optimistic entrepreneur dismisses it. Because noise is so high, they attribute the poor performance entirely to bad luck (\(\epsilon_{1}\)) rather than a fundamental flaw in the idea (\(\theta\)). They get trapped in a zombie project, thinking, “This signal is just garbage noise; I know my underlying idea is brilliant, so I’m pushing through.”

Given the (subjective) optimal decision, it is feasible to find the chance that the entrepreneur would push/pivot relative to decision made by an unbiased decision-maker. In low noise regimes, when information is clear, an optimistic entrepreneur never mistakenly pushes through on a fundamentally weak project. Their systematic bias manifests entirely as an Optimism Flight Error (Type I Error), abandoning a viable project due to the inflated perceived value of their next outside option.

In contrast, in noisy environments, the entrepreneur treats negative feedback as mere random noise rather than an indicator of low underlying project productivity. Here, the entrepreneur never prematurely flees a project; instead, they suffer exclusively from an Optimism Trap Error (Type II Error), continuing to waste effort pushing a project that objective reality dictates should be terminated.

Similar to the above analysis, it is feasible to examine the consequences of pessimism, i.e., \(\mu_{e}<\mu\). Interestingly, we find identical qualitative behaviors with an important difference - Pessimism Trap occurring in low noise regimes where they push through projects because of their lower perceived value of their next outside option, and Pessimism Flight occurring in high noise regimes where they see a negative signal and take it as definitive proof of failure.

Table below summarizes the behavioral consequences of optimism and pessimism in different information environments.

So far, we have characterized how the decisions are affected by the pessimism or optimism of the entrepreneur. While no one chooses to be optimistic or pessimistic (!), it is instructive to look at the consequences of the same degree of optimism or pessimism. To formally evaluate the economic consequences of miscalibrated entrepreneurial beliefs, we establish the objective ex-ante expected payoff function, denoted by \(\Pi(K^{*})\), as a function of an arbitrary decision threshold \(K^{*}\).

If the first-period performance signal satisfies \(K<K^{*}\), the entrepreneur pivots to a new idea and receives the objective expected baseline payoff \(E[\hat{\theta}+\epsilon_{3}]=\mu\). Conversely, if \(K\ge K^{*}\), the entrepreneur pushes through with the current project, yielding a true expected cumulative payoff of \(E[2\theta+\epsilon_{2}\mid K]=2E[\theta\mid K]\).

Under objective reality, the fundamental quality is distributed as \(\theta\sim N(\mu,\sigma^{2})\) and the first-period noise is \(\epsilon_{1}\sim N(0,\tau^{2})\). Thus, the observed signal is distributed as \(K\sim N(\mu,\Omega^{2})\), where \(\Omega=\sqrt{\sigma^{2}+\tau^{2}}\). Standard Bayesian updating yields the true conditional expectation of project quality given the performance signal:

The objective ex-ante expected payoff \(\Pi(K^{*})\) is the weighted average of these outcomes across the true distribution of the signal \(K\):

where \(f(K)\) represents the probability density function (PDF) of \(K\sim N(\mu,\Omega^{2})\). Utilizing the standard property of normal distributions \(\int^{\infty}_{K^{*}}(K-\mu)f(K)\,dK=\Omega\phi\left(\frac{K^{*}-\mu}{\Omega}\right)\), this expression simplifies to:

where \(\Phi(\cdot)\) and \(\phi(\cdot)\) denote the standard normal cumulative distribution function (CDF) and PDF, respectively.

To verify that perfectly calibrated beliefs (\(\mu_{e}=\mu\)) maximize the true expected economic value, we differentiate \(\Pi(K^{*})\) with respect to the threshold \(K^{*}\). Defining the standardized threshold variable \(z=\frac{K^{*}-\mu}{\Omega}\) and noting that \(\frac{d\phi(z)}{dz}=-z\phi(z)\), the first-order condition is given by:

Setting the first derivative to zero yields the unique critical point \(K_{\text{obj}}\):

Substituting \(\Omega^{2}=\sigma^{2}+\tau^{2}\) simplifies this exactly to the objective optimal decision threshold:

Since \(\frac{d\Pi(K^{*})}{dK^{*}}>0\) for \(K^{*}<K_{\text{obj}}\) and \(\frac{d\Pi(K^{*})}{dK^{*}}<0\) for \(K^{*}>K_{\text{obj}}\), the threshold \(K_{\text{obj}}\) constitutes a unique global maximum.

To evaluate the relative economic damage of overconfidence versus underconfidence, we examine the third derivative of the payoff function evaluated at the optimal threshold \(K_{\text{obj}}\):

We analyze the economic loss via a third-order Taylor expansion of the loss function \(\text{Loss}(K^{*})=\Pi(K_{\text{obj}})-\Pi(K^{*})\) around the optimal benchmark. For an arbitrary directional perturbation \(x>0\):

Because \(\Pi''(K_{\text{obj}})<0\) and \(\Pi'''(K_{\text{obj}})<0\), both terms in the expansion for a positive threshold deviation are strictly positive. This establishes a structural asymmetry principle:

This mathematical inequality provides the structural foundation for our comparative analysis across information environments:

• Low-Noise Environment (\(\sigma>\tau\)): The signal multiplier \(\left(1-\frac{\tau^{2}}{\sigma^{2}}\right)\) is positive, meaning an optimist (\(\mu_{e}=\mu+\Delta\)) sets a positive threshold deviation (\(K_{\text{opt}}=K_{\text{obj}}+x\)) while a pessimist (\(\mu_{e}=\mu-\Delta\)) sets a negative threshold deviation (\(K_{\text{pess}}=K_{\text{obj}}-x\)). Consequently, \(\text{Loss}(\text{Optimist})>\text{Loss}(\text{Pessimist})\).

• High-Noise Environment (\(\tau>\sigma\)): The signal multiplier \(\left(1-\frac{\tau^{2}}{\sigma^{2}}\right)\) is negative, flipping the directional impact of beliefs. The optimist sets a negative threshold deviation (\(K_{\text{opt}}=K_{\text{obj}}-x\)) while the pessimist sets a positive threshold deviation (\(K_{\text{pess}}=K_{\text{obj}}+x\)). Consequently, \(\text{Loss}(\text{Pessimist})>\text{Loss}(\text{Optimist})\).

Summarizing the key insights, we obtain the following: In mature, low-noise markets—such as opening a fast-food franchise or a traditional retail branch—early performance data is highly predictive; here, optimism leads to a devastating “Zombie Trap,” causing founders to ignore clear signals and waste capital on unviable projects. However, in novel, high-noise frontier sectors—such as biotechnology, deep tech, or unformed digital platforms—early feedback is heavily obscured by random noise. In these volatile environments, an even slightly pessimistic decision-maker suffers from defeatist flight, abandoning a potentially revolutionary breakthrough at the first minor setback. In contrast, the optimistic entrepreneur possesses the exact psychological insulation required to endure messy signals, serving as an evolutionary safeguard against premature abandonment. Ultimately, our framework reveals that entrepreneurial optimism is not a universal cognitive flaw, but an ecologically rational feature vital for sustaining innovation in the world’s most uncertain environments.