Spatial externalities and risk in interdependent security games


Individuals regularly invest in self-protection to reduce the risk of an adverse event. The effectiveness of self-protection often depends on the actions of other economic agents and can be modeled as a stochastic coordination game with multiple Pareto-ranked equilibria. We use lab experiments to analyze tacit coordination in stochastic games with two kinds of interdependencies in payoffs: “non-spatial” in which every agent’s action has an impact on the risk faced by every other agent, and “spatial” in which agents only impact the risk faced by their immediate neighbors. We also compare behavior in the stochastic games to deterministic versions of the same games. We find that coordination on the payoff-dominant equilibrium is significantly easier in the deterministic games than in the stochastic games and that spatial interdependencies lead to greater levels of coordination in the deterministic game but not in the stochastic game. The difficulty with coordination observed in the stochastic games has important implications for many real-world examples of interdependent security and also illustrates the importance of not relying on data from deterministic experiments to analyze behavior in settings with risk.

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  1. 1.

    Other early examples of such coordination games include Cooper et al. (1990) and Van Huyck et al. (1991). See also the survey article by Devetag and Ortmann (2007).

  2. 2.

    For other related work, see Berninghaus et al. (2002) and Corbae and Duffy (2008).

  3. 3.

    Another research question concerns the speed with which subjects converge to an equilibrium. Cassar (2007) hypothesizes that a shorter characteristic path length — the average number of shortest paths of one subject to all other subjects — causes faster convergence to any equilibrium. In our experimental set-up, the characteristic path length in the non-spatial treatment is just 1 (everybody is connected directly with everybody else) while it is 1.8 in the spatial treatment.

  4. 4.

    Kunreuther et al. (2009) analyze behavior in IDS games with strategic substitutability, stochastic versions of the classic prisoner’s dilemma. These games have only one equilibrium, which is sub-optimal. For example, the equilibrium number of vaccinations is likely to be less than the optimal number since the private benefit of a vaccine is decreasing as the number of people vaccinated increases.

  5. 5.

    As usual in these coordination games, there is also a Nash equilibrium in mixed strategies in each game: all players take action with probability 0.6, which is Pareto-inferior to everybody taking action with certainty (the slight asymmetry in expected payoffs in the spatial game, with 28 instead of 30 in Table 2, was chosen in order to have the same NE in mixed strategies in both games).

  6. 6.

    Technically, the random number was not determined in the session itself; rather, all 900 numbers (18 subjects in a session times 50 rounds) were randomly determined before the first session in order to avoid behavioral differences between sessions due to differences in how bad outcomes were generated. This use of a “pseudo-random feed” has been standard in lab experiments with random effects; see, for example, Cox et al. (2001).

  7. 7.

    Of course, many of those subjects did not suffer the bad outcome themselves but were able to observe on the payoff screen that others in their group did, which might have influenced them negatively.

  8. 8.

    This result is consistent with findings for public good games, where cooperation in situations with stochastic losses tends to be lower than in their deterministic counterparts (Berger and Hershey 1994).


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Correspondence to Stephan Kroll.

Additional information

This research was supported in part by funds provided by the Rocky Mountain Research Station, Forest Service, U.S. Department of Agriculture.

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Kroll, S., Shafran, A.P. Spatial externalities and risk in interdependent security games. J Risk Uncertain 56, 237–257 (2018).

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  • Self-protection
  • Risk mitigation
  • Experiments
  • Interdependent security
  • Coordination game

JEL Classifications

  • D81
  • C91