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When can a predictive model improve by anticipating behavioral reactions to its predictions?

This is Jessica. Most of my research involves data interfaces in some way or another, and recently I’ve felt pulled toward asking more theoretical questions about what effects interfaces can or should have in different settings. For instance, the title of the post is one question I’ve started thinking about: In situations where a statistical model is used to predict human behavior, and people have access to the predictions, under what conditions can we expect the model to perform better when it explicitly tries to estimate how people will react to the displayed predictions. By explicitly I mean estimating behavioral reactions to the display is part of the model specification. 

The answer depends of course on how one formalizes it (how you define a data generating process, restrict the space of possible models to be explored, define the strategies and payoffs of agents, etc.). But I think it’s an interesting thought exercise.

When might we want to ask such a question? One situation that comes to mind is election forecasting, where you have many people looking at predictions of election outcomes (vote share per candidate for instance) created by news agencies or pollsters etc. Sometimes there are concerns that people will “best respond” to the predictions in ways that change the outcome of the election. For example, presidential election forecasting involves predicting both vote choice and turnout, where turnout might be affected by perceived closeness of the race, i.e., the less close the election, the less mobilized some voters. The effect of the display on behavior here might be thought of as somewhat accidental; people want to know the future, perhaps in part to decide how to act, but not necessarily. Hence there’s a demand for forecasts, but the fact that their availability might change the election outcome in any significant way is perceived as a nuisance or risk. A beneficent forecaster might care about possible behavioral reactions to the display because they would like for their forecast to reduce the amount of regret that eligible voters feel post election over their choice of whether to vote and what candidate to vote for.

There are many other situations where interfaces serve up predictions with the goal of directly informing behavior, e.g., recommender systems. For instance, apps for driving directions like Google Maps predict current travel times along different routes, and the developer might naturally want the predictions to achieve some aggregate goal, like less congested traffic. 

Recently, I did some work with Paula Kayongo, a Ph.D student I work with, and Jason Hartline, a game theorist, which is partially what got me thinking about this question. We came up with the notion of a visualization equilibrium: the visualization of predicted behavior such that when you show people that visualization you observe that behavior. We used a congestion game set up to test whether visualizing a Nash equilibrium for the game leads to that outcome realizing. Not surprisingly, it doesn’t. People have various strategies they use to react to the display such that you can see a different outcome than what was predicted.

In our work so far, the visualization is simply a reflection of past plays of the game, which can be thought of as a simple form of model prediction. But this is less realistic than a setting where the display presents the predictions of some statistical model which might be informed by past behavior but not identical to it. Often, if we think the format of the interface has an effect on decisions people make from it, we might do some “offline” experiments to try to find the version that leads to the least bias and just use that. But if people are reacting to the content of the prediction, it might be worth trying to learn those dynamics as part of the model. So I started wondering, if you have a model that predicts behavior, and you expect people might try to best respond to the visualized predictions in ways that can change the outcome, under what circumstances should you try to anticipate (model) these dynamics directly? 

At a high level, we can think of the display effect as a mapping between a visualization of some predicted outcome to a realized outcome. We can think of a predictive model that anticipates reactions to its own predictions as one that tries to estimate this function.

There are a bunch of parameters to define to pose the question more rigorously. If we assume we have a model that makes predictions about behavior in some payoff-relevant state, where the behavior of others impacts the utility of different possible actions a person faces, then we should consider:

  • What are the parameters of the “game”, including the action set available to agents, payoff functions, assumptions about agents (e.g., economic rationality), etc.?
  • Are all states payoff relevant states, or just some/one of them? (e.g., a political election is a one-shot scenario, but using google maps on a trip is not). 
  • When are predictions available to the agents? Continuously or at certain time points?  
  • What’s the relationship between those who see the display and those who will act in the payoff-relevant state? Is it the same group, or is the former a subset? 
  • How is the space of possible models constrained? What’s the functional form used to estimate the display effect? What form do the data inputs available to the model at any given point take?
  • Where does the state(s) for which the question is being asked occur in a process of best response dynamics? In other words, how far is the system from equilibrium? 

To answer the question requires constraining it by deciding these parameters, but I find the idea of formalizing it and then working out the answer compelling as a thought exercise. Maybe we’ll get more insight into real world interface dynamics. Even without deciding on a specific case to study, we can make conjectures like, modeling display effects is unlikely to be helpful when a system is in equilibrium, or when those who view the predictions make up only a small portion of those who acts in the payoff-relevant state. It also seems related to the martingale property, since if its a martingale your forecast at any time point should be an unbiased predictor of the forecast at any later time. If you expect a lot of movement in the predictions in the future, your uncertainty must be high, hence you won’t be “surprised” by some reaction to the display. There’s lots more that could be said, but for now I’ll just pose the high level question.


  1. Ben says:

    > where the behavior of others impacts the utility of different possible actions a person faces

    The one person version of this seems interesting enough. Any particular reason for the multiple person version?

    • I think of it as inherently about strategic behavior. E.g., if I didn’t see an election forecast making me pretty certain my candidate would win, I might have gone and voted. So my choice of strategy must depend on what I think other people will do.

      • Ben says:

        Oh okay. But can’t this sort of prediction thing be relevant even without a cost function?

        For example, there’s a poll, and someone publishes an election forecast. Everyone then adjusts their predictions of the outcome according to the poll.

        Then there’s another poll, and another election forecast gets published, and everyone adjusts their predictions again.

        • Reminds me of some the discussion here: eg of mean reversion in polls. I agree there are pieces that can be analyzed purely statistically. To me the election forecast example implies a cost function if we assume the forecast is modeling both choice and turnout like most of the major ones do, since turnout is only a concern because there’s a “cost” to voting. I’m not sure what the best way to model opinion alone would be, and how displays might affect that. Do undecided voters become more likely to choose in favor of the candidate they perceive to be winning?

  2. Sandro Ambuehl says:

    Your question is closely related to a core topic in economics, the Lucas Critique, and the ensuing rational expectations revolution (see The details are very different, but the foundations, and likely some of the tools might prove quite useful.

    • Thanks! Part of asking this question is me trying to figure out what pieces of econ and statistical theory to look to, and I wasn’t aware of Lucas Critique. It does seem pretty related.

      Unrelatedly, though I’m not an economist, I’m a fan of your work. I read Belief Updating and the Demand for Information awhile back with some of my PhD students and spent time thinking about it.

    • Dale Lehman says:

      Another important area where some research in economics may have been done (I don’t follow this literature, so I can’t provide any references, but perhaps someone else, Sandro?, can): The Federal Reserve has always been concerned with how people will respond to their announcements and predictions. I’m fairly sure that their models, or at least what they publicly reveal about their models, takes into account the behavioral response they believe it will provoke. This is one of the reasons why Milton Friedman had always favored a passive FED – a narrowly defined scope of behavior with relatively little discretion, since a more activist FED would bring about reactions that would undermine whatever the FED intended.

      There is probably some similar literature in finance: earnings announcements and other corporate disclosures are likely to bring about reactions, so where there is any discretion about these announcements, the models must be taking into account the potential responses.

      The visualization application you suggest seems quite interesting and amenable to experimentation. I like the idea of a Nash equilibrium where the response to the visualization is what the visualization intended to provoke.

      • Michael J says:

        I haven’t studied this either but the way it was taught to me in macro 101 was that the Fed solved this issue by being abundantly clear about what they intend so the market is never surprised. Like a big problem before was inflation would chase inflationary expectations i.e. there would be a feedback loop of expecting inflation to increase soon so you raise prices which causes inflation itself which then shifts expectations further up and so on and so forth. So then after the Volcker era that tamped down inflation to a reasonable amount, the Fed has really really cared about anchoring inflationary expectations by being predictable and transparent. In that way I think the Fed can be both activist and not succumb to that Friedman criticism.

        Anyways, this was just to add some context to the idea about the Fed being concerned about market response to their announcements.

      • Yes, I had a hunch that finance might be a place where this is relevant … I spoke at a conference organized by the Fed and Swiss National Bank a few years ago and realized that financial theory is relevant to a lot of my interests in uncertainty and modeling. Didier Sornette also sent me a few papers related to this (eg and mentioned it’s related to the efficient market hypothesis.

        Glad you like the equilibrium idea!

  3. Kenneth Tay says:

    This paper (Perdomo et al. 2020) from the ML community calls it performative prediction, and there are some references in the Related Work section:

  4. Rick G says:

    I have a game I’ve played with students where each is asked to pick a real number between 0 and 100 that will be closest to half of the mean pick for the classroom. We play this iterative over class sessions, so each student is aware of previous winning picks. I don’t know what the equilibrium looks like or if there is one, or how one might model it, but is sure is fun.

  5. Bill Harris says:

    Jessica, you sound as if you’re about to develop the field of system dynamics, as Jay Forrester and his group developed it at MIT starting in the 1950s and ’60s. It’s a way to understand the cause of and solve problems, mostly in “managed systems”: systems that have as one part the actions of a person or people who, for better or worse, seek to control (managed) the behavior of the system.

    Suggestions for reading? Perhaps today’s canonical text is John Sterman’s /Business Dynamics/. Jay Forrester’s early /Industrial Dynamics/ still has much to offer. Tom Fiddaman, who occasionally comments here, has a blog on the subject at

    SD models are sets of simultaneous nonlinear ODEs. Vensim, STELLA, and Powersim are generally the big three SD simulators, but, if you can code an ODE, that’s sufficient; the SD simulators just make the process much easier. I used MCSim, which some here may know, for several years because I found it offered some pretty nice benefits.

    There are a lot more references to suggest, but the easiest way might be to check out and skim a copy of /Business Dynamics/. Besides the general content, it lists many references. You could also try some of Tom Fiddaman’s Model Library (Vensim has a free Model Reader) that will likely run almost all of those models.

    • Bill Harris says:

      The key contribution of system dynamics is the application of the mindset of the feedback control engineer to managed systems, even though much of (at least early) feedback control theory was focused on linear or linearized systems, and most managed systems seem inherently nonlinear.

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