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Erschienen in: Public Choice 3-4/2019

05.10.2018

The attack and defense of weakest-link networks

verfasst von: Dan Kovenock, Brian Roberson, Roman M. Sheremeta

Erschienen in: Public Choice | Ausgabe 3-4/2019

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Abstract

We experimentally test the qualitatively different equilibrium predictions of two theoretical models of attack and defense of a weakest-link network of targets. In such a network, the attacker’s objective is to assault at least one target successfully and the defender’s objective is to defend all targets. The models differ in how the conflict at each target is modeled—specifically, the lottery and auction contest success functions (CSFs). Consistent with equilibrium in the auction CSF model, attackers utilize a stochastic “guerrilla-warfare” strategy, which involves attacking at most one target arbitrarily with a random level of force. Inconsistent with equilibrium in the lottery CSF model, attackers use the “guerrilla-warfare” strategy and assault only one target instead of the equilibrium “complete-coverage” strategy that attacks all targets. Consistent with equilibrium in both models, as the attacker’s valuation increases, the average resource expenditure, the probability of winning, and the average payoff increase (decrease) for the attacker (defender).

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Fußnoten
1
Applications of the weakest-link structure include: organizational performance that depends on the weakest-link (Kremer 1993); internet security (Moore et al. 2009); and package auctions in which the objective of some bidders is to obtain all of the goods while for other bidders the objective is to obtain only one good (Milgrom 2007).
 
2
As stated in the Joint House-Senate Intelligence Inquiry into September 11, 2001 (US Congress 2002), terrorists need to be successful only once in killing Americans and demonstrating the inherent vulnerabilities the victims face.
 
3
However, as noted in the case of a single two-player contest with linear costs—by Baye et al. (1994) and Alcalde and Dahm (2010)—there exist equilibria that are payoff equivalent to the \(r = \infty\) case whenever \(r > 2\). Ewerhart (2017) has demonstrated that, in fact, in such an environment any Nash equilibrium is payoff and revenue equivalent to the all-pay auction. Thus, the auction CSF case with \({\text{r}} = \infty\) is a relevant theoretical benchmark for all \(r > 2\).
 
4
For the attacker, this prediction holds for all parameter configurations. For the defender, this prediction holds if the ratio of the attacker’s valuation of success to the defender’s valuation of success is below a certain threshold.
 
5
See also the related papers Dziubiński and Goyal (2013, 2017).
 
6
For almost all configurations of the players’ valuations of winning, one of the two players drops out with positive probability by allocating zero resources to each target, with the identity of the dropout determined by a measure of asymmetry in the conflict that takes into account both the ratio of the players’ valuations and the number of targets.
 
7
For a survey, see Kovenock and Roberson (2012). Recent theoretical work on multi-battle/Blotto-type games includes extensions such as: asymmetric players (Roberson 2006; Hart 2008; Weinstein 2012; Dziubiński 2013; Macdonell and Mastronardi 2015), non-constant-sum variations (Szentes and Rosenthal 2003; Kvasov 2007; Hortala-Vallve and Llorente-Saguer 2010, 2012; Roberson and Kvasov 2012; Ewerhart 2018), alternative definitions of success (Golman and Page 2009; Tang et al. 2010; Rinott et al. 2012), and political economy applications (Laslier 2002; Laslier and Picard 2002; Roberson 2008; Bierbrauer and Boyer 2016; Boyer et al. 2017; Thomas 2018). An extensive theoretical literature also exists on dynamic multi-battle contests. See, for instance, Harris and Vickers (1987), Klumpp and Polborn (2006), Konrad and Kovenock (2009) and Gelder (2014).
 
8
The experimental literature on dynamic multi-battle contests also is growing. See, for instance, Deck and Sheremeta (2012, 2015), Mago et al. (2013), Mago and Sheremeta (2017, 2018) and Gelder and Kovenock (2017).
 
9
For the auction CSF game, if the players allocate the same level of the resource to a target, it is assumed that the defender wins the target. However, a range of tie-breaking rules yields similar results. A detailed description of the theoretical model can be found in Clark and Konrad (2007) for the lottery CSF and Kovenock and Roberson (2018) for the auction CSF.
 
10
For example, players were called participants and targets were called boxes.
 
11
We chose these parameter valuations to ensure that: (a) the four treatments cover both parts of both propositions, and (b) each subject faced a non-trivial allocation problem in which both the attacker and the defender had a substantial chance of winning some targets.
 
12
Risk aversion preferences also were elicited, along the lines of Holt and Laury (2002). We found no interesting patterns between risk attitudes and behavior in weakest-link contests and omit discussion of this issue.
 
13
Role-switching avoids any social preferences, i.e., subjects who were assigned as disadvantaged attackers knew that they would also play the role of the advantaged defenders, and induces better learning, since subjects have an opportunity to learn game strategies in both roles.
 
14
When both players allocated the same amount to a given target, the computer selected the defender as the winner of that target.
 
15
Although the comparative statics results are framed in terms of a change in the attacker's valuation, owing to invariance of preferences with respect to affine transformations of utility, the theoretical benchmark also would apply to a decrease in the unit cost of resource expenditure of the attacker.
 
16
A standard Wald test, conducted on estimates of panel regression models, rejects the hypothesis that the average expenditures under the lottery CSF are equal to the predicted theoretical values in Table 3 (all p values < 0.05). Under the auction CSF we can reject the null hypothesis only for the defender (p value < 0.05).
 
17
For a detailed review of possible explanations for the over-expenditure phenomenon see Sheremeta (2013, 2016).
 
18
We combined the distribution of tokens to each of the four targets into one target, since marginal distributions to each target are identical across targets.
 
19
Note that our results are robust to the exclusion of strategies involving drastic over-expenditure. For example, if defining drastic over-expenditure as expenditure greater than twice the expected Nash equilibrium expenditure (which excludes 18% [13%] of attacker strategies in the Lottery-Low [Lottery-High] treatment), then excluding drastic over expenditure the attacker allocates resources to at most one target 60% of the time in the Lottery-Low treatment and 57% of the time in the Lottery-High treatment. The attacker allocates a positive amount to all four targets 19% of time in the Lottery-Low treatment and 26% in the Lottery-High treatment.
 
20
Attacker strategies involving drastic over-expenditure have smaller expected payoffs than those without and tend to be positively correlated with the number of targets receiving a positive allocation. For example, in the Lottery-Low (Lottery-High) treatment 49% (72%) of all of the attacker strategies involving drastic over-expenditure are four-target strategies. In looking at the entire sample in the Lottery-Low (Lottery-High) treatment, the average expected payoff of the observed attacker strategies that allocate resources to only one target is 2.84 (23.11) and the average expected payoff of the observed attacker strategies that allocate resources to all targets is 3.13 (22.83).
 
21
If we exclude strategies involving drastic over-expenditure, namely 11% (27%) of the defender strategies in the Lottery-Low (Lottery-High) treatment, the defender allocates resources to all targets 92% in the Lottery-Low treatment and 79% in the Lottery-High treatment. In the latter treatment, no targets receive a positive allocation 17% of the time.
 
22
In calculating the empirical mass points at 0 (Figs. 1, 2), we use an allocation of less than one token as an approximation of 0. That approximation is adopted because the tie-breaking rule favors defenders and therefore may encourage attackers to place a very small allocation on some targets in order to reduce the tie-breaking disadvantage. However, even if we use only 0 allocations to compute mass points at 0, we still get results that are close to the theoretical predictions (in the Auction-Low and Auction-High treatments, for example, the mass points at 0 for the attackers are 0.6 and 0.5).
 
23
If we exclude strategies involving drastic over-expenditure, which excludes 18% (16%) of attacker strategies in the Auction-Low (Auction-High) treatment, then the attacker allocates resources to at most one target 87% of the time in the Auction-Low treatment and 79% of the time in the Auction-High treatment.
 
24
The fact that the defender allocates 0 resources to all four targets 30% of the time could, potentially, be explained by subjects changing role assignments after 10 periods and a period 1–10 attacker continuing to behave as an attacker during periods 11–20. However, our results are robust to restricting data to the first 10 periods. Furthermore, excluding strategies involving drastic over-expenditure, which excludes 5% (20%) of defender strategies in the Auction-Low (Auction-High) treatment, the defender allocates resources to all targets 87% of the time in the Auction-Low treatment and 54% of the time in the Auction-High treatment. In the latter treatment, no targets receive a positive allocation 37% of the time.
 
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Metadaten
Titel
The attack and defense of weakest-link networks
verfasst von
Dan Kovenock
Brian Roberson
Roman M. Sheremeta
Publikationsdatum
05.10.2018
Verlag
Springer US
Erschienen in
Public Choice / Ausgabe 3-4/2019
Print ISSN: 0048-5829
Elektronische ISSN: 1573-7101
DOI
https://doi.org/10.1007/s11127-018-0618-1

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