Identifying Strategic Effects in Observed Choices

Why Co-Movement Does Not Reveal Strategy

When two firms, households, or other decision-makers choose similar actions, it is tempting to conclude that one responds strategically to the other. But both choices may instead respond to the same market condition: a demand boom, a regulatory change, or an unobserved local preference. The central identification problem is to separate strategic responses from shared causes of observed choices.

This distinction matters for prediction and policy. A model that attributes co-movement to strategic interaction may predict that changing one player’s incentives will affect others. If co-movement actually comes from common shocks, that prediction can be wrong. Identification therefore requires variation that shifts a player’s incentives in a way that is not itself a direct payoff shift for the other player.

A Simple Structural Model

Consider two players \(i\) and \(j\) choosing continuous actions \(y_{im}\) and \(y_{jm}\) in market \(m\). A local linear representation of player \(i\)’s best response is

y_{im} = a_i + \beta_i x_{im} + \gamma_i y_{jm} + \omega_m + \varepsilon_{im}.

Here \(x_{im}\) is an observed payoff shifter for player \(i\), \(\omega_m\) is a market-level shock shared by players in market \(m\), and \(\varepsilon_{im}\) is a player-specific unobserved payoff component. The coefficient \(\gamma_i\) is the slope of player \(i\)’s best response to the other player’s action in this specification. The model is an illustrative linear, interior-choice model; it is not a universal representation of every strategic setting.

Definition: Actions are strategic complements for player \(i\) when an increase in the other player’s action raises \(i\)’s best response; in the linear model, this means \(\gamma_i > 0\). They are strategic substitutes when an increase in the other player’s action lowers \(i\)’s best response; here, \(\gamma_i < 0\). If \(\gamma_i=0\), the best response in this model does not depend directly on the other player’s action.

The structural coefficient is not generally the same as the observed association between choices. If \(\omega_m\) raises both players’ choices, their actions may be positively associated even when \(\gamma_i=0\). Conversely, if player-specific unobserved factors affect both choices, their association may reflect correlated unobservables rather than a causal response. In the equation above, \(y_{jm}\) is an equilibrium choice, so it can be correlated with the unobserved components in player \(i\)’s equation.

Worked Example: The Same Correlation, Different Explanations

Suppose choices have been centered around their means. In one hypothetical model, there is no strategic interaction: \(y_1=\omega+\varepsilon_1\) and \(y_2=\omega+\varepsilon_2\). Let the common shock have variance \(4\), and let the independent player-specific shocks each have variance \(1\).

Then \(\operatorname{Cov}(y_1,y_2)=4\), while \(\operatorname{Var}(y_1)=\operatorname{Var}(y_2)=4+1=5\). The correlation is

\operatorname{Corr}(y_1,y_2) = \frac{4}{\sqrt{5\cdot 5}} = 0.8.

Now consider a different hypothetical model with no common shock: \(y_1=0.5y_2+v_1\) and \(y_2=0.5y_1+v_2\), where \(v_1\) and \(v_2\) are independent with variance \(1\). Solving the two equations gives \(y_1=(v_1+0.5v_2)/0.75\) and \(y_2=(0.5v_1+v_2)/0.75\). Their variances are both \(1.25/0.5625=2.222\), and their covariance is \(1/0.5625=1.778\). Thus their correlation is \(1.778/2.222=0.8\).

The same observed correlation, \(0.8\), is consistent with a common-shock explanation and with a strategic-complementarity explanation. Correlation by itself does not distinguish them.

Using an Excluded Payoff Shifter

A common identification strategy uses a variable \(z_{jm}\) that shifts player \(j\)’s payoff, but does not directly shift player \(i\)’s payoff. Such a variable is excluded from player \(i\)’s structural equation. If \(z_{jm}\) changes player \(j\)’s choice and player \(i\) responds to that choice, it can provide information about the strategic effect.

For this interpretation to be credible, several conditions are needed. The shifter must change player \(j\)’s choice after accounting for the included controls (relevance). It must not directly affect player \(i\)’s payoff or action except through the strategic effect being studied (exclusion). It must also be unrelated, conditional on those controls, to the unobserved determinants of player \(i\)’s choice, including the common market shock (exogeneity). The design additionally requires that the observed response is not confounded by a change in market composition or another simultaneous market event. These are substantive assumptions about how the shifter is generated, not consequences of fitting a regression.

The logic can be seen in a symmetric two-player version of the model. Let both players’ strategic slope be \(\gamma\), and let \(z_j\) enter only player \(j\)’s payoff with coefficient \(\beta\). Suppose \(|\gamma|<1\), so the linear system has a unique solution and the equilibrium comparative statics below are well-defined. Hold other shifters and shocks fixed. A one-unit increase in \(z_j\) changes the equilibrium choices according to

\frac{\partial y_i}{\partial z_j} = \gamma\frac{\partial y_j}{\partial z_j}, \qquad \frac{\partial y_j}{\partial z_j} = \beta+\gamma\frac{\partial y_i}{\partial z_j}.

Substituting the first expression into the second gives \((1-\gamma^2)\frac{\partial y_j}{\partial z_j}=\beta\). Therefore,

\frac{\partial y_j}{\partial z_j}=\frac{\beta}{1-\gamma^2}, \qquad \frac{\partial y_i}{\partial z_j}=\frac{\gamma\beta}{1-\gamma^2}.

Provided \(\beta\neq 0\), the response ratio is \(\frac{\partial y_i/\partial z_j}{\partial y_j/\partial z_j}=\gamma\). The excluded shifter first changes the player whose payoff it shifts; any response by the other player is transmitted through the strategic link in this model. The ratio is a population comparative-static result under the stated assumptions. In empirical work, it is estimated from variation across observations, with uncertainty and controls accounted for.

Worked Example: Reading the Responses to an Excluded Shifter

Suppose a hypothetical policy shifter \(z_j\), measured in policy units, affects only player \(j\)’s payoff. The model’s direct response coefficient is \(\beta=2\) choice units per policy unit, and the equilibrium response of player \(j\) is estimated as \(2.381\) choice units per policy unit. Under the symmetric linear model, the strategic slope is the response ratio.

Using \(1-\gamma^2=\beta/(\partial y_j/\partial z_j)\), the estimates imply \(1-\gamma^2=2/2.381\approx0.8400\), so \(\gamma^2\approx0.1600\). The response of player \(i\) determines the sign. If it is \(0.9524\) choice units per policy unit, then

\widehat{\gamma} = \frac{0.9524}{2.381} \approx 0.4000.

The positive ratio is consistent with strategic complements in this model: player \(i\)’s choice rises when the excluded shifter raises player \(j\)’s choice. The result is conditional on the exclusion and exogeneity assumptions; the arithmetic alone does not establish them.

Market Variation and Common Shocks

Markets can provide useful variation when shifters differ across markets while the strategic relationship is stable. For example, a cost shifter may vary for one player across locations for reasons unrelated to local demand. If its changes are plausibly independent of common market conditions, they can help separate the response transmitted through the other player’s action from the response to a shared demand shock.

Market fixed effects can remove a common additive component \(\omega_m\) when it is constant within each market and there is suitable within-market variation in the variables of interest. But adding fixed effects does not automatically make an endogenous choice exogenous, nor does it repair a shifter that directly affects both players. Identification still rests on a credible source of variation that satisfies the relevant exclusion and exogeneity conditions. If the shifter varies only across markets, absorbing market fixed effects may also absorb that variation, leaving no identifying information.

Worked Example: A Common Shifter Is Not an Excluded Instrument

Consider a hypothetical market-level demand shifter \(z\) that increases both players’ payoffs directly. Let the symmetric strategic slope be \(\gamma=0.4\), and suppose the direct effect of \(z\) on each player’s choice equation is \(\beta=2\). A one-unit increase in \(z\) changes each choice by the same amount \(r\), since the players are symmetric. The equilibrium equation for either player gives \(r=2+0.4r\), so

r = \frac{2}{1-0.4} = 3.333.

Both choices therefore rise by \(3.333\) choice units per unit of \(z\), and their response ratio is \(3.333/3.333=1\). That ratio does not recover the true strategic slope of \(0.4\). The reason is that \(z\) directly shifts both players’ choices, violating the exclusion restriction. A high correlation or similar response across players would not, on its own, show that strategic effects are large.

What the Design Can and Cannot Establish

A credible excluded-shifter design identifies a strategic effect only relative to its model and assumptions. In particular, a researcher must defend why the shifter is relevant for the targeted player, why it has no direct effect on the other player, and why its variation is not associated with unobserved market conditions after conditioning on appropriate controls. The first stage should be examined: a theoretically appealing shifter that barely moves the targeted choice provides little empirical information about the response ratio.

The source of variation also matters. If markets differ in institutions or demand in ways correlated with the shifter, cross-market comparisons can mix strategic responses with omitted market differences. If the strategic slope varies across markets, a single average response ratio may not describe every market. And if multiple equilibria are possible, observed choices may depend on which equilibrium is realized; in that setting, the simple comparative statics above need not describe the observed response without further assumptions.

The goal is not to label every correlated choice as strategic or to treat every cost or policy variable as a valid instrument. It is to identify a source of variation that changes one player’s incentives, trace how the other player responds, and assess whether competing explanations—especially common shocks—are ruled out by the design.

Common Mistakes

Reading a positive correlation as complementarity. A common market shock can raise both choices. The correction is to identify variation that isolates a strategic response rather than relying on raw co-movement.
Calling a variable “excluded” because it is absent from a regression. Exclusion is a claim about the underlying payoff and causal pathways. A variable that directly affects the other player’s payoff is not excluded just because it was omitted from that equation.
Ignoring common shocks after finding an instrument. The instrument must also be exogenous to unobserved determinants of the response equation. If it tracks market demand, the resulting estimate can still be confounded.
Confusing a reduced-form response with the strategic slope. A shifter can affect a player’s choice through equilibrium feedback. In the symmetric linear model, the strategic slope is the ratio of responses, not generally either response on its own.

Key takeaway: Observed choices can move together because players respond to one another or because they share common shocks. A strategic interpretation requires credible, relevant payoff variation excluded from the other player’s direct payoff, together with assumptions that rule out alternative paths from the shifter to observed choices.

Check Your Understanding

Use the model and identification conditions developed above.

Why can a positive correlation between two players’ choices occur even when their choices are strategic substitutes?
In the symmetric linear model, what does a negative response ratio imply, and which assumptions are needed to interpret it that way?
Give one reason a player-specific cost shifter might fail the exclusion restriction for the other player.
When can market fixed effects remove a common shock, and why might they also remove the variation needed for identification?
If an instrument strongly shifts one player’s choice but is correlated with unobserved market demand, which identification condition fails?