Abstract
Contemporary theories of deterrence place a strong emphasis on coherency between model and theory. Schelling’s contention of irrational threats for successful deterrence abandons the rationality assumption to explain how a player can deter, thereby departing from the standard game theoretic solution concepts. It is a misfit model in relation to a deterrence theory and, therefore, excluded. The article defends and remodels Schelling’s intuition by employing the level-k model. It is shown that an unsophisticated player that randomizes over its strategies brings about an advantageous outcome. The model also shows that the belief that a player randomizes has the same deterrent effect, as an actual stochastic choice, like Schelling suggested. While this means Schelling’s idea can be saved, it is still problematic how we should view contributions of bounded rationality in relation to current deterrence theory. The article suggests that separating the purpose of a model in conjunction with allowing other scientific ideals than model-theory coherence permits a broader and philosophically sounder approach.
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Notes
This ideal is captured by van Fraassen’s criteria that “A model is called a model of a theory exactly if the theory is entirely true if considered with respect to the model alone” (van Fraassen, 1989, 218).
Schelling discusses various ways to interpret and understand Chicken, including historically tracing it back to its first account in the Iliad.
Schelling discusses the behaviour in a mental ward as well as using the literary example form Joseph Conrad’s antagonist in The Secret Agent. The mental ward patients represent both types of strategy: genuine insanity and faked. The Conrad example is used to describe the shrewd strategist. See also Zagare’s discussion 2018, 433; and Field’s comments 2014, 10.
Zagare (2018) addresses Schelling’s solution explicitly, and Zagare and Kilgour (2000, 54) categorise it as one of the deterrence paradoxes,“threats that leaves something to chance” (Paradox II). This is also Zagare and Kilgour’s critique of Powell’s Nuclear Deterrence analysis, which builds upon Schelling’s idea but expands it to a Bayesian game.
This conclusion is not unique to Zagare and Kilgour, Powell (1990), de Mesquita (1992) and Slantchev (2011) view play in sequence as an important part for correct representation. However, Selten’s Entry Deterrence game and sub-game perfect equilibrium was not developed at the time when Schelling made his contributions—although he discusses games in strategic form where he assumes that one player moves first.
A naïve interpretation of mixed strategies NE is that players randomize between their choices when the product of the payoffs and the probabilities are equal between outcomes. This interpretation can be sufficient in some cases. However, if we want to represent strategically savvy players such as the US and USSR, this type of analysis may leave us wanting. There are two alternative interpretations. One view is that randomization represents a lack of knowledge for the players (Rubenstein, 1991). According to this interpretation, the players are insecure of which strategy to choose, and, given the payoffs, tends towards cooperation, but may defect to a degree. A second interpretation suggested by Aumann and Brandenburger is that the probability distribution should be understood as a degree of belief in a given strategy (Aumann & Brandenburger, 1992). This means that the players are willing to play defect in the cases when they believe that this will not result in Conflict (DD), but refrain from such choices in other cases.
One can take a different tack all together, Alexander Field point out that it is odd that Schelling’s solution has been accepted to the degree that it has. To Field it is part of an argument for why game theory is a faulty method for strategic analysis of deterrence (Field, 2014). Zagare (2018) show that field’s analysis partly misses the point as Field has not considered the advances of Perfect Deterrence, which does exactly what Field calls for, namely model credible threats with the method Field distrusts—game theory.
For instance, the discussions about Iraq and North Korea bear the mark of some these strategic deliberations (see for instance Allison, 2010; Salik, 2014; van de Velde, 2010 regarding discussion of proliferation and future scenarios and Lambertini (2013) for a game theoretic analysis of North Korean development of nuclear arms).
The so-called perfect asymmetric deterrence game, analyses Challenger–Defender interaction in extensive form. This game have been statistically tested by Quackenbush with good results (Quackenbush, 2010, 2011a). The study does not discriminate between convectional and nuclear deterrence. However, since nuclear deterrence never has had a conflict outcome (like conventional deterrence), the result, while important is not conclusive regarding nuclear deterrence. See also Rauchhaus (2009) who investigates the nuclear peace hypothesis, and Narang (2013) who studies nuclear postures.
Haruvy and Stahl (2005) tested which selection criteria people seemed to rely on when playing a game for the first time. They compare (1) payoff-dominance, when players choices focuses on which outcome yields the most, (2) risk-dominance when players choose outcomes where the uncertainty is the lowest, (3) maximin, when players the seek out the outcome where they maximize the minimum amount that they can get and level-1 bounded rationality, where players randomizes over their possible strategies (see also Stahl (1993) for the original model; Camerer (2003) for a general overview).
The arbitrary nature of type 0 is sometimes understood as not an existing real type, only as a fictional type in the mind of a type 1 player.
The analysis holds as long as \(p \ne 1/2\). A special case is when \(p = 1/2\). In such cases both players will randomize over their strategy spaces. Type 0, because it always randomizes over its strategy space and type \(k \ge 1\) because it is its best reply to this strategy when \(p=1/2\) (type 1 because it is the best reply to type 0; type 2 because it is the best reply to type 1; type 3 because it is the best reply to type 2).
Here we assume that the level of capability is sufficient so that Status Quo (CC) is preferred over Conflict (DD), which is in keeping with the definition of capability by Zagare, Kilgour and Quackenbush.
The Semantic View maintains that a scientific theory is the collection of models within it. Central is therefore coherence between model and theory. This philosophical position is an answer to the Syntactic View that aims to recast the mathematical expression of a model to a meta-mathematical language like predicate logic.
References
Alaoui, L., Janezic, K. A., & Penta, A. (2021). Reasoning about others’ reasoning. Journal of Economic Theory, 189, 1–51.
Allison, G. (2010). Nuclear disorder: Surveying atomic threats. Foreign Affairs, 89(1), 74–85.
Arad, A., & Rubinstein, A. (2012). The 11–20 money request game: A level-kReasoning study. American Economic Review, 102(7), 3561–3573.
Aumann, R., & Brandenburger, A. (1992). Epistemic conditions for Nash equilibrium. Econometrica, 63(5), 1161–1180.
Binmore, K. (2009). Rational decisions. Princeton University Press.
Brams, S., & Kilgour, M. (1988). Game Theory and National Security. Blackwell Publishing.
Camerer, C. F. (2003). Behavioural game theory: Experiments in strategic interaction. Princeton University Press.
Carlson, L., & Dacey, R. (2006). Sequential analysis of deterrence games with a declining status quo. Conflict Management and Peace Science, 23(2), 181–198.
Cartwright, N. (1983). How the laws of physics lie. Oxford University Press.
Clarke, K. A., & Primo, D. M. (2007). Modernizing political science: A model-based approach. Perspectives on Politics, 5(4), 741–753.
Craver, C. F. (2002). Structures of scientific theories. In P. K. Machamer & M. Silberstein (Eds.), Blackwell guide to the philosophy of science (pp. 55–79). Blackwell.
Crawford, V. P. (2003). Lying for strategic advantage: Rational and boundedly rational misrepresentation of intentions. The American Economic Review, 93(1), 133–149.
Crawford, V. P., & Costa-Gomes, M. A. (2013). Structural models of nonequilibrium strategic thinking: Theory, evidence and applications. Journal of Economic Literature, 51(1), 5–62.
de Mesquita, B. B., & Lalman, D. (1992). War and reason: Domestic international imperatives. Yale University Press.
Field, A. J. (2014). Schelling, von Neumann, and the event that didn’t occur. Games, 5(1), 53–89.
Gigerenzer, G. (2002). The adaptive toolbox. In R. Selten & G. Gigerenzer (Eds.), Bounded rationality: The adaptive tool box (pp. 37–50). MIT Press.
Grüne-Yanoff, T., & Schweinzer, P. (2008). The role of stories in applying game theory. Journal of Economic Theory, 15(2), 131–146.
Haruvy, E., & Stahl, D. O. (2005). Equilibrium selection and bounded rationality in symmetric normal-form games. Journal of Economic Behaviour and Organization, 62(1), 98–119.
Johnson, J. (2014). Models amongst political scientists. American Journal of Political Science, 58(3), 547–560.
Kahn, H. (1960). On thermonuclear war. Princeton University Press (Reprinted by Transaction Publishers 2007).
Lambertini, L. (2013). Joining the Nuke Club: A forward induction approach. Defence and Peace Economics, 24(1), 15–21.
Luce, D. R., & Raiffa, H. (1957). Games and decisions: Introduction and critical survey. Reprint 1989. Dover Publications.
Narang, V. (2013). What does it take to deter? Regional nuclear postures and international conflict. Journal of Conflict Resolution, 57(3), 478–508.
Powell, R. (1990). Nuclear deterrence: The search for credibility. Cambridge University Press.
Quackenbush, S. L. (2004). The rationality of rational choice theory. International Interactions, 30(2), 87–105.
Quackenbush, S. L. (2010). General deterrence and international conflict: Testing perfect deterrence theory. International interactions, 36(1), 60–85.
Quackenbush, S. L. (2011). Understanding general deterrence: Theory and application. Palgrave MacMillan.
Quackenbush, S. L. (2011). Deterrence theory: Where do we stand? Review of International Studies, 37(2), 741–762.
Quackenbush, S., & Zagare, F. (2016). Modern deterrence theory: Research trends, policy debates and methodological controversies. Oxford Handbooks Online.
Rauchhaus, R. (2009). Evaluating the nuclear peace hypothesis: A quantitative approach. Journal of Conflict Resolution, 53(2), 258–277.
Rubenstein, A. (1991). On the interpretation of game theory. Econometrica, 59(4), 909–924.
Russell, B. (1959). Common sense and nuclear warfare. Simon and Schuster.
Salik, N. (2014). Nuclear terrorism: Assessing the danger. Strategic Analysis, 28(2), 173–184.
Schelling, T. C. (1960). The strategy of conflict. Harvard University Press.
Schelling, T. C. (1966). Arms and influence. Yale University Press.
Selten, R. (1978). The chain store paradox. Theory and Decision, 9(2), 127–159.
Slantchev, B. L. (2011). Military threats: The costs of coercion and the price of peace. Cambridge University Press.
Sörenson, K. (2017). Comparable deterrence: Target, criteria and purpose. Defense Studies, 17(2), 198–213.
Stahl, D. O. (1993). Evolution of smart n players. Games and Economic Behavior, 5, 604–617.
van de Velde, J. R. (2010). The impossible challenge of deterring ‘Nuclear Terrorism’ by Al Qaeda. Studies in Conflict and Terrorism, 33(8), 682–699.
van Fraassen, B. (1989). Laws and symmetry. Oxford University Press.
Walt, S. (1999). Rigor or rigor mortis? Rational choice and security studies. International Security, 23(4), 5–48.
Zagare, F. (2018). Explaining the long peace: Why von Neumann (and Schelling) got it wrong. International Studies Review, 20(3), 422–437.
Zagare, F., & Kilgour, M. (1993). Asymmetric deterrence. International Studies Quarterly, 37(1), 1–27.
Zagare, F., & Kilgour, M. (2000). Perfect deterrence. Cambridge University Press.
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Sörenson, K. A Misfit model: irrational deterrence and bounded rationality. Theory Decis 94, 575–591 (2023). https://doi.org/10.1007/s11238-022-09907-7
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DOI: https://doi.org/10.1007/s11238-022-09907-7