Abstract
Thomas Schelling (Arms and influence, Yale University Press, New Haven, 1966) cites bridge burning as a method of commitment. While such a commitment can increase the chances of success in a conflict, it generally will lower one’s payoff if the conflict is lost. I use a contest framework and establish conditions under which such commitment can raise a player’s expected payoff. The comparative static effects of bridge burning never are favorable at an interior equilibrium, but the strategy may induce the opponent to concede the contest’s outcome. I also analyze the strategy, associated with Sun Tzu, of leaving an escape path open for one’s enemy. This strategy always succeeds at an interior equilibrium and raises the expected payoffs of both players. Under certain parameter restrictions, leaving an open escape path also has the potential of inducing the opponent to concede the contest. A special case of the model is offered to explain why a group vulnerable to a wealth transfer might prefer a less efficient tax system to a more efficient system.
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Notes
Later Muhammad Ali. The quote is reproduced in Roberts and Smith (2016, p. 43).
Bridge burning is referenced in The Strategy of Conflict but not addressed in detail as in Arms and Influence. Another instance wherein Schelling discusses commitment strategies is the battle for self-control. See chapters 5–8 in Schelling (2006).
The idea is related to gambling for resurrection, whereby a leader facing the loss of power is willing to undertake risky military actions because victory is the only way of retaining power. See Downs and Rocke (1994).
Bridge burning also has similarities to entry deterrence in the industrial organization literature. In his discussion of entry deterrence, Tirole (1988, p. 316) provides an informal discussion of bridge burning within in the context of a military example. As in the current paper, in Tirole’s informal example, bridge burning succeeds by discouraging an attack.
Thus, the restriction on r allows us to avoid a lengthy discussion about what happens when both constraints are violated. Equilibria for r > 2 in a symmetric contest have been characterized by Alcade and Dahm (2010) and Ewerhart (2015). They are mixed strategy equilibria. The equilibrium outcomes bear a strong resemblance to the outcomes of the all-pay auction. Analyses of the all-pay auction may be found in Hillman and Samet (1987), Hillman and Riley (1989) and Baye et al. (1996). Ewerhart (2015, pp. 67–70) suggests that similar results will apply for asymmetric contests.
A recent example that considers negative prizes is Liu et al. (2018). They consider negative prizes as part of an optimal contest design to induce maximal effort on the part of the participants.
Nti (1999) analyzes contests wherein players’ valuations of payoffs are asymmetric.
The constraints do not take into account the fixed costs ε, which are assumed to be close to zero. Rather they simply require that, at the time the contest is to be held, each player earns an expected payoff exceeding \(- {\kern 1pt} {\kern 1pt} R_{i}^{L}\) at the interior Nash solution reflected in Eqs. (5a, 5b). Technically, however, interior Nash equilibria exist wherein a player’s payoff exceeds \(- R_{i}^{L}\), but by less than ε, so that he would strictly prefer to concede the game at stage 2. We ignore that possibility because ε is close to zero, but it could be an issue if \(R_{2}^{L}\) can rise without bound. See footnote 19.
It is important to note that raising \(R_{2}^{L}\) increases player 2’s loss when she loses the contest, but also increases her loss when she concedes the contest. Thus, an increase in \(R_{2}^{L}\) does not make it less likely that player 2’s participation constraint is satisfied. In fact, just the opposite holds because the increase in relative stakes makes it more likely that she would prefer to participate in the contest.
In light of the substitution a = (r − 1)Rr, r < 1 implies a < 0, which is not permissible. A negative a would, however, make the right-hand side of (9) negative, explaining why the inequality would hold when r < 1.
Suppose that player 1’s participation constraint is satisfied, so that a > (r − 1)Rr, and that player 2’s constraint initially is violated. Further suppose that player 2 can raise \(R_{2}^{L}\) sufficiently such that her participation constraint is satisfied, but that she cannot raise it sufficiently such that player 1’s constraint is violated. Would she want to burn bridges under that scenario? The answer is no. Initially, she prefers conceding the contest to participating so that the payoff at the interior equilibrium must be less than the initial value of \(- R_{2}^{L}\). Our prior analysis shows that player 2’s payoffs at the interior equilibrium fall with increases in \(R_{2}^{L}\), when a > (r − 1)Rr. Thus, the payoff at the interior equilibrium that player 2 induces by increasing \(R_{2}^{L}\) will be less than her payoff if she conceded the contest initially at the original value of \(R_{2}^{L}\).
Suppose that initially player 2’s participation constraint is violated and player 1’s is not. Would player 2 want to lower \(R_{1}^{L}\) in order to allow her participation constraint to be satisfied? The answer is yes. Since \(R_{2}^{L}\) is constant in that scenario, player 2 can cause her constraint to hold only if she raises her payoff above the fixed value -\(R_{2}^{L}\) at the interior equilibrium and doing so unambiguously makes her better off. The just-stated result contrasts with bridge burning, wherein player 2 is increasing \(R_{2}^{L}\) and will make herself worse off even though changes in \(R_{2}^{L}\) can induce her participation constraint to hold. See footnote 14.
Nothing of substance is changed if we assume the minimum value lies above zero, though a minimum value above zero would affect the precise expressions presented below in Result 5.
Because the analysis indicates that player 2 never will utilize the bridge burning strategy at an interior equilibrium, I have not presented the derivative showing the effect of such a strategy on player 1. However, it can be shown that such a strategy would lower player 1’s expected payoff. Thus, if it were invoked at an interior equilibrium, bridge burning would make both players worse off.
Recall that we are assuming that r > 1.
In the statement of the game, it is assumed that \(\overline{{R_{2}^{L} }}\) is finite. If \(R_{2}^{L}\) can be increased without bound, then bridge burning can succeed even if r < 1. From (5a), if R can be increased without bound, then at some point we will have \(\prod_{1\,} - \,\varepsilon < - \,R_{1}^{L}\). At that point, player 1 would prefer to concede at stage 2 rather than proceed to the contest. In assuming that \(\overline{{R_{2}^{L} }}\) is finite and that ε is near zero, I am assuming implicitly that such an outcome cannot emerge in the game I have specified. If we allow that fixed costs ε could be large or that \(R_{2}^{L}\) could become infinite, then that case would become relevant, expanding the circumstances under which bridge burning would be effective.
The transfer game as specified involves only two players. If we have a group 1 with n1 identical players and a group 2 with n2 identical players, then the transfer paid by each member of group 2 in the event of a loss would be \(R_{2}^{L} = (1 + \delta )R_{1}^{W} n_{1} /n_{2}\). Adding the constant n1/n2 to the model would not affect any of its conclusions. With many members, each group would be vulnerable to a free-rider problem. When there are no income effects, as in the current model, within each group only the player with the largest stakes in the contest will contribute. If players are identical, aggregate contributions from the group are equivalent to the contributions a single individual would make. The results are consistent with Olson’s (1965) exploitation of the great by the small result as well as his group-size paradox. See the discussions of those issues in Sandler (2015), Buchhotz and Sandler (2016) and Pecorino (2015, 2016). Consideration of a free-rider problem would not affect the conclusions discussed in the main text.
Mulligan also quotes from Fischer and Summers (1989, p. 387) who write, “Thus, a better tax system may lead to more wasteful spending, better cars to more speeding, and better inflation protection to more inflation.”.
The Military Institutions of the Romans, Book 3, available at http://www.digitalattic.org/home/war/vegetius/index.php#b319.
The strategy of leaving an escape path open will be robust to small reductions in one’s own payoff in the event of victory. It is only when that reduction is sufficiently large that the strategy will become undesirable. Thus, if bringing a dictator to justice is a second-order concern relative to getting him to give up power, leaving the open escape path (i.e., allowing for asylum) will be optimal.
See, for example, MacDonald’s (2012) account of the Cuban missile crisis.
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Acknowledgements
I would like to thank B.C. Kim, two anonymous referees, seminar participants at Auburn University and participants at the Public Choice Society and American Economic Association Meetings for providing helpful comments on this paper.
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Pecorino, P. Bridge burning and escape routes. Public Choice 184, 399–414 (2020). https://doi.org/10.1007/s11127-019-00726-z
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DOI: https://doi.org/10.1007/s11127-019-00726-z