Defending against multiple different attackers Kjell Hausken, Vicki M. Bier

1. Defending against multiple different attackersKjell Hausken, Vicki M. Bier Advisor: Yeong-Sung Lin Presented by I-Ju Shih

2. Agenda • Introduction • The model • Analysis of the simultaneous game • Two-period game when defender moves first • Two-period game when attackers move first • Comparing the three games • Conclusions

3. Agenda • Introduction • The model • Analysis of the simultaneous game • Two-period game when defender moves first • Two-period game when attackers move first • Comparing the three games • Conclusions

4. Introduction • A great deal of research in the past decade has been concerned with conflicts between a single defender and a single attacker. • Bier (2004) lays out the general rationale for a game-theoretic approach to such conflicts. • Some recent work considers multiple attackers. • Contests with multiple attackers can credibly be analyzed as rent-seeking models.

5. Introduction • One defender defends, and multiple heterogeneous attackers attack, an asset. • This paper uses game theory and rent-seeking model for the interactions among the defender and the attackers. • This paper models two such differences here-differences in the values they ascribe to the defended asset, and differences in their unit cost of attack. • Most past literature has focused on sequential games in which the defender moves, this paper also considers games in which the defender moves simultaneously with the attackers, and games in which the attackers act first, leaving the defender to move second.

6. Agenda • Introduction • The model • Analysis of the simultaneous game • Two-period game when defender moves first • Two-period game when attackers move first • Comparing the three games • Conclusions

7. The model

8. The model • The basic model • Players: This paper considers a game with one defender and n attackers, which gives a set of n + 1 agents. The agents compete for a valuable asset controlled by the defender. • Strategies: Each attacker has the option of launching an attack, with the goal of acquiring a portion of the defender’s asset. Achieving a level of attack effort Ti is assumed to require an attack expenditure Fi(Ti), where əFi/əTi > 0. The defensive expenditure required to achieve t is f(t), where əf/ət > 0. For simplicity, f = ct and Fi = CiTi.

9. The model • The basic model • Contest success functions: The contest between the defender and the attackers for the asset is assumed to take the common ratio form frequently used in the rent-seeking literature.

10. The model • The basic model • Payoffs: The profits (payoffs or utility functions) u(t,T1, . . . ,Tn) of the defender and Ui(t,T1, . . . ,Tn) of attacker i are given by • Sequence of play: Three scenarios are considered: the agents move simultaneously; the defender moves first; or the attackers move first.

11. Agenda • Introduction • The model • Analysis of the simultaneous game • Two-period game when defender moves first • Two-period game when attackers move first • Comparing the three games • Conclusions

12. Analysis of the simultaneous game • General analysis • This paper solves for Nash equilibrium, defined such that no agent can benefit by unilaterally deviating from his equilibrium strategy. • The first-order and second-order conditions for an interior solution are • Solving for the interior solution gives

13. Analysis of the simultaneous game • General analysis • Proposition 1. Conditions for deterrence in the simultaneous game: The defender gives up its asset if Attacker i ceases attacking if • Proof. Follows from requiring t and Ti, respectively, to be non-negative in Eq. (4). When t>0, . When t<=0,

14. Analysis of the simultaneous game • General analysis • The profits of the n + 1 agents are given by

15. Analysis of the simultaneous game • Special cases of the simultaneous game(oneattacker) • C/R = c/(k r), which means that the defender is disadvantaged with a ratio c/r that is k times as high as C/R for the attacker. When k > 1, the defender is disadvantaged. When 0 < k < 1, the attacker is disadvantaged. When k = 1, both agents are equally advantaged. • Inserting n = 1 and C/R = c/(k r) in Eqs. (4) and (5) gives R=r

16. Analysis of the simultaneous game • Special cases of the simultaneous game(n attackers) • Ci/Ri = C/R = c/(k r). • Substituting into Eqs. (4) and (5) gives k>1, n>=2

17. Analysis of the simultaneous game • Special cases of the simultaneous game(n attackers) • Proposition 2. Conditions for deterrence in the simultaneous game with homogeneous attackers: Assume that all attackers have equivalent characteristics, Ci/Ri = C/R=c/(kr). (a) If the defender is disadvantaged with k>=n/(n-1), then the defender ceases investing, gives up the asset, and earns zero profit. (b) The attackers always attack. • Proof. Follows from requiring t<=0 in Eq. (7), and noting that Ti > 0 in Eq. (7).

18. Analysis of the simultaneous game • Special cases of the simultaneous game(n attackers) k>=n/(n-1)

19. Analysis of the simultaneous game • Corner solutions of the simultaneous game when two agents remain(one defender and one attacker) • Attacker i ceases attacking if • When one attacker is removed from the game in this manner, the parameter n is reduced by one, and the resulting (revised) inequality for each of the remaining attackers is reevaluated. • Assume there are only n = 2 attackers left, i and j. If Ci/Ri>=c/r + Cj/Rj, then attacker i ceases attacking. This means that only attacker j remains to compete with the defender. • Solving the first-order conditions for the defender and attacker j gives

20. Analysis of the simultaneous game • Corner solutions of the simultaneous game when two agents remain(two attackers) • The defender gives up its asset if • Solving the first-order conditions for the n attackers gives

21. Analysis of the simultaneous game • Corner solutions of the simultaneous game when two agents remain • Proposition 3. Conditions for deterrence in the simultaneous game when two agents remain: (a) When one attacker and one defender remain in the contest, the attacker does not withdraw regardless of how large Ci/Ri is, and the defender also does not withdraw. (b) When c/r is high, so that the defender withdraws, and two attackers remain, they never withdraw.

22. Analysis of the simultaneous game • Corner solutions of the simultaneous game when two agents remain • Proof. Proposition 3(a) follows from Eq. (8), which shows that setting Ti = 0 leads to Tj > 0 and t > 0. Proposition 3(b) follows from Eq. (10), where setting t = 0 implies that Ti > 0 and Tj > 0. Contest success functions

23. Analysis of the simultaneous game • Numerical examples for the simultaneous game • For simplicity, C1/R1 = 1. c/r+C2/R2>1 c/r+C2/R2<1

24. Analysis of the simultaneous game • Numerical examples for the simultaneous game • c/r = C1/R1 = r = Ri = 1 for i = 1, 2.

25. Analysis of the simultaneous game • Numerical examples for the simultaneous game • c/r = 1.5 and C1/R1 = 0.5, r = Ri = 1 for i = 1, 2.

26. Agenda • Introduction • The model • Analysis of the simultaneous game • Two-period game when defender moves first • Two-period game when attackers move first • Comparing the three games • Conclusions

27. Two-period game when defender moves first • General analysis • This paper solves for a sub-game perfect equilibrium defined as follows: A strategy profile is a subgame perfect equilibrium if it represents Nash equilibrium of every subgame of the original game. • A common method for determining subgame perfect equilibria in the case of a finite game is backward induction, which means that the second period is solved first, followed by the solution of the first period.

28. Two-period game when defender moves first • General analysis • The first-order and second-order conditions of the n attackers for the second period are as given in Eq. (3), and imply the following results: • To derive the new first-order and second-order conditions, we begin by summing up T1 + T2 +…+ Tn for the n attackers, yielding

29. Two-period game when defender moves first • General analysis • Solving for the sum T1 + T2 +…+ Tn then yields • Inserting (13) into the denominator in (2) and simplifying gives the defender’s first-period profit as follows:

30. Two-period game when defender moves first • General analysis • Differentiating u with respect to t to determine the defender’s first-order and second-order conditions for an interior solution gives

31. Two-period game when defender moves first • General analysis

32. Two-period game when defender moves first • General analysis • Proposition 4. Conditions for deterrence in the sequential game when the defender moves first: (a) The defender gives up its asset, and is deterred from investing in defense, if (b) Attacker i ceases attacking if • Proof. Follows from Eqs. (15) and (17), and the fact that t and Ti must be nonnegative.

33. Two-period game when defender moves first • Special cases when defender moves first(oneattacker) • Letting n = 1 and C/R = c/(k r) in Eqs. (15), (17), and (18) gives When k > 1, the attacker has a second-mover disadvantage compared with the simultaneous game. When k <= 1/2, the attacker does not attack, and earns zero profit.

34. Two-period game when defender moves first • Special cases when defender moves first(nattackers) • Letting Ci/Ri =C/R = c/(k r) in Eqs. (15), (17), and (18) gives

35. Two-period game when defender moves first • Special cases when defender moves first(nattackers) • Proposition 5. Equilibrium solutions for the sequential game with homogeneous attackers and defender when the defender moves first: When C/R = c/r (so k = 1) and r = Ri, the defender invests as much as all n attackers taken together (t/Ti = n), and earns as much profit as all n attackers taken together (u/Ui = n). The defender’s profit is inversely proportional to n, while each attacker’s profit is inversely proportional to n2. • Proof. Follows from Eq. (20) when k = 1.

36. Two-period game when defender moves first • Special cases when defender moves first(nattackers) • Proposition 6. Conditions for deterrence in the sequential game with homogeneous attackers when the defender moves first: Assume Ci/Ri = C/R. (a) If the defender is disadvantaged with k>=n/(n-1), then the defender ceases investing, gives up the asset, and earns zero profit. (b) The attackers will all withdraw whenever k <= n/(n + 1). • Proof. Follows from Eq. (20), and the fact that t and the Ti are non-negative.

37. Two-period game when defender moves first • Special cases when defender moves first(nattackers) k>=n/(n-1) k<=n/(n+1) n/(n-1)<k<n/(n+1)

38. Two-period game when defender moves first • For simplicity, C1/R1 = 1.

39. Two-period game when defender moves first • c/r = C1/R1 = r = Ri = 1 for i = 1, 2.

40. Two-period game when defender moves first • c/r = 1.5 and C1/R1 = 0.5, r = Ri = 1 for i = 1, 2.

41. Agenda • Introduction • The model • Analysis of the simultaneous game • Two-period game when defender moves first • Two-period game when attackers move first • Comparing the three games • Conclusions

42. Two-period game when attackers move first • General analysis • The defender’s first-order condition for the second period is given by the first equation in (3). • The first-period profits of attacker i as • The first-order conditions for an interior solution gives

43. Two-period game when attackers move first • General analysis • Summing up the Ti for the n attackers gives • which can be solved to yield • Inserting Eq. (25) into Eqs. (21) and (23) gives

44. Two-period game when attackers move first • General analysis • Proposition 7. Conditions for deterrence in the sequential game when the attackers move first: The defender gives up its asset if Attacker i ceases attacking if • Proof. Follows from Eq. (26), the non-negativity of t and the Ti. • In the simultaneous game and the defender moves first: The defender gives up its asset if

45. Two-period game when attackers move first • Special cases when attackers move first(oneattacker) • n = 1 and C/R = c/(k r) into Eqs. (26) and (27) gives When k > 1, the (single) attacker invests more than the defender does, and earns a higher profit. When k >=2, the defender is so disadvantaged that it ceases defending.

46. Two-period game when attackers move first • Special cases when attackers move first(nattackers) • Let Ci/Ri = C/R = c/(k r). In this case, Eqs. (26) and (27) give • Proposition 8. Equilibrium solutions for the sequential game when the attackers move first: When k=1, then Ti/t = (2n-1)/n, Ui/u = (2n-1)/n. The defender’s profit is inversely proportional to n2, while the profit of any given attacker decreases more slowly than 1/n2 when there are multiple attackers (since 2n-1 > n when n > 1). • Proof. Follows from Eq. (29) when k = 1.

47. Two-period game when attackers move first • Special cases when attackers move first(nattackers) • Proposition 9. Conditions for deterrence in the sequential game with homogeneous attackers when the attackers move first: Assume Ci/Ri = C/R. If k>=2n/(2n-1), then the defender ceases investing, gives up the asset, and earns zero profit, and the attackers never withdraw. • Proof. Follows from Eq. (29), and the fact that t and Ti must be non-negative. • In the defender moves first game: If k>=n/(n-1), then the defender ceases investing, gives up the asset, and earns zero profit.

48. Two-period game when attackers move first • Special cases when attackers move first(nattackers) k>=2n/(2n-1)

49. Two-period game when attackers move first • Special cases when attackers move first • C1/R1 = 1.

50. Two-period game when attackers move first • Special cases when attackers move first • c/r = C1/R1 = r = Ri = 1 for i = 1, 2.