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Computer Science

Procedia Computer Science 12 (2012) 404 - 411

Complex Adaptive Systems, Publication 2 Cihan H. Dagli, Editor in Chief Conference Organized by Missouri University of Science and Technology

2012- Washington D.C.

An approximation algorithm for computing a tipping set in super modular games for interdependent security

B. Cremeans, S. Lakshmivarahan*,S.K. Dhall

School of Computer Science, University of Oklahoma, Norman, OK, 73072

Abstract

The problem of finding the minimal tipping set in a super modular game is known to be NP-hard. In this paper, we derive an approximation algorithm to find a minimal tipping set. In the special case of the uniform game, the approximation provides the exact result.

Keywords:Tipping; Game Theory;

1. Introduction

With ever growing dependency through commerce and communication between various parts of the globe, there is a greater need for the analysis and understanding of interdependent security (IS) among interacting agents. IS has been successfully modeled using the framework of n-person (non-cooperative) game theory [1]. In a series of seminal papers [2][3][4], Heal and Kunreuther (hereafter H-K) describe these models and their applications to airline security as well as vaccination games to prevent the spread of infectious diseases, etc. In particular, the models are used for airline baggage security policy adoption. The special features of this model include the following: (a) each player is endowed with only two pure strategies or actions, invest - (1), and not invest - (0) in security measures; (b) the players are not allowed to use randomized strategies which in turn implies that we are interested only in the Nash equilibria (NE) in pure strategies [5]; (c) the utility (negative of the cost) function for each player has two components, first is due to one's own action (1 or 0) and the second is due to the action (1 or 0) of the other players. This second component is called the externality component, which in turn decides the level of

* Corresponding author. Tel.: +1-405-325-2978; fax: +0-000-000-0000 .

1877-0509 © 2012 Published by Elsevier B.V. Selection and/or peer-review under responsibility of Missouri University of Science and Technology. doi:10.1016/j.procs.2012.09.094

interdependency among the players; and (d) a subset of players acting in collusion, by the clever choice of their own actions, can influence the externalities of other players not in the coalition so as to force them to change their choice of actions. This phenomenon, whereby one subset can exert influence over another is called tipping [6][7] or cascading [8].

A condition for an n-person game to exhibit the tipping / cascading property is that the payoff (loss) function of the players must satisfy the increasing (decreasing) differences property. This property is intimately associated with the super-modularity of the utility functions [3][9] In this paper we work with the differences in the losses, which are the negation of the payoff differences.

Super-modular functions and functions with increasing differences are defined on lattices [9]. In our case, the underlying binary lattice is defined by the 2n binary strings of 1 's and 0's under the natural partial order defined over the binary strings. It turns out this binary lattice is also a binary hypercube of dimension n. See Figure 1a for an example of a four dimensional lattice which is also a four dimensional binary cube.

H-K were the first to analyze the tipping phenomenon in the context of interdependent security games that arise in the context of airline security games [2][4]. By concentrating on n-airline security games with two NE - one at 0" and one at V they proved the existence of a minimal tipping set. However, it turns out that the problem of finding a minimal tipping set is combinatorially difficult [4] and no viable algorithm is yet known. This difficulty is a result of (n-1)! different paths connecting the NE at 0" with that at 1". In this paper we propose an approximation algorithm for finding a locally minimal tipping set. Our method exploits the topological properties of the underlying binary hypercube by enumerating the O(n) node disjoint path distribution in the binary lattice which is also a binary hypercube.[10]

The basic game model is given in Section 2. The complexity of the problem of finding the minimal tipping set is described in section 3. Section 4 provides an overview of our approximation algorithm's candidate path selection. An example is contained in section 5 and an algebraic method to calculate a minimal tipping set from the candidate sets is described in Section 6. The special case of uniform games is analyzed in Section 7 and another example provided. A short summary of the paper is given in Section 8.

Figure 1: (a)State Lattice ^Interdependence Graph

Table 1: 3 Player Losses

Play\Cost Player 1 Player 2 Player 3

000 P1+ L + (1 - A)*(q,1 + q31 - q,1q31 )L P2 + L, + (1 - p,)*(q12 + q32 - q ,q3, )L, p3 + L3 + (1 - p3)*(q 3 + q23 - qnq23 )Ln

001 P1L1 +(1 - p0* q?1L1 p?L? +(1 - p,)* q1,L, C 3 ^ (q13 ^ q23 _ q13q23)L3

010 P1L1 +(1 - P0* q31L1 C2 ^ (q12 ^ q32 _ q1?q3?)L? p3L3 + (1 - P3)* q3L3

011 P1L1 c 2 1 2 C 3 1 qnL3

100 c1 + (q71 + q31 - q21q31)L1 p2L2 +(1" p? ) * q32L2 T?3L3 +(1- p3)* q23L3

101 C1 + q21L1 p2L2 C 3 1 q23L3

110 C1 + q31L1 c 2 1 2 T?3L3

111 C1 C2 C3

2 The Game Model

There are n players labeled 1 through n, each endowed with two pure strategies denoted by 1 (invest) and 0 ( not invest). A play s is defined by the n-tuple s = (s1,s2,...,Sn) where st £ {0,1} denotes the choice of pure strategy

by player l<i<n. There are a total of 2n distinct plays denoted by S = {s | S = (s1,s2,...,Sn),s. £ {0,1}}. For a,bGS, define a binary relation < as follows. Clearly 0<1. We say a<b (or b>a ) when a.<b.ior l<i<n. It can be

verified that the pair (S,<) is a poset and is indeed a complete lattice[9]. Let u.s^R where

u(s) = (u(1 s),U2(s),...,Un(s)) denote the n-tuple of utility functions with ur.S^R denoting the utility of the

player i. The game is specified by (S,u). Let s_. denote a play (s1,...,st 1,*,si+1,...,sn) and (s_..,1..) denote the play

(s1,..., s. 1,1i, si+1,..., sn ). The game (S,u) is super-modular[9] if, for every i,

u,(s' .,1.)-u,(s' .,0.)>u.(s .,1.)-u.(s .,0.)

;v —i r ;v —i r ;v —i r ;v —i r

when s,_>s_.. Intuitively, this decreasing difference condition states that the loss for player i to change from

strategy 0 to 1 does not increase when a subset of the other players has already moved from 0 to 1.

We now define the airline security game which is predicated on the natural assumption that a player can die no more than once [2]. The utility, or the pay-off, is defined in terms of losses. Let c>0 be the cost of investment

(choice of strategy 1) in security by player i, L>0 be the cost or loss due to a catastrophic incident, p>0 be the

probability that player i will suffer a catastrophic loss due to his own inaction (choice of strategy 0) and q..

(0<q ..<1) be the probability that player j will suffer a catastrophic loss due to inaction of player i. For later i.

reference, define an n*n matrix Q=[q-] with q..=0. Clearly, the off-diagonal elements of Q define the

interdependency among the n players. It can be shown [4] that where u. is the average cost due to self action and (2)

u. is the average cost due to the action of others, the total expected cost is given by:

u.(s)=u(1\s)+u(2\.s)

ui1)(s)=s ic i+(1~s L i

(2) (3)

u(2) = (1 - (1 - Si)pt)(1 -n,v,(l - (1 - s,)qii))Li (4)

where n^Lia^a^-a^refers to the product of the a.. An example of a three person game is given in Table 1.

3 Complexity of Tipping in a Super modular game

Analysis of tipping is concerned with the difference

u i(s_i\i)-ui(s_pi)

where (s_.) is an edge in the complete lattice considered as a binary hypercube. A sequence of differences

along a path in the binary hypercube connecting the NE at 0 to the one at 1 is given by

ui(0n"11i)-ui(0n"10i)>ui(0n"21i1j.)-ui(0n"21i0/.)>...>ui(1n"11i)-ui(1n"10i) (4)

Since 0n and 1n are NE, clearly ui(0n~l1i)~ui(0n~l0i)>0 and ui(1n_11i)-ui(1n_10i)<0. That is, the above

sequence of decreasing real numbers starts from a positive value and ends up at a negative value. Thus there is a zero crossing point that corresponds to a subset (kj,...,ki} players choosing pure strategy 1. But there are (n-1)! distinct sequences corresponding to (n-1)! disjoint paths in the underlying hypercube. Hence, we get (n-1)! subsets, one for each path. The minimal tipping set is then to be derived from those sets. Hence, the problem of finding the minimal tipping set is NP-hard.

HK proved that for the special case satisfying two conditions (player and state independence assumptions) there exists a minimal tipping set. Recently, Cremeans et. al. [11] gave a simpler algorithm to find such a tipping set for this special case.

In this paper, our goal is to find an approximation to the minimal tipping set for the general case.

4 An Approximation algorithm

We can exploit the lattice / hypercube structure of the game states examined in [12] to sample possible solutions. Recall that a given decreasing sequence of inequalities of the differences in the losses uniquely induces a mapping of the sequences in (4) onto edges in the binary hypercube of dimension n as follows:

1 2 n -1

(0n-11i,0n-10i)-> (0n-%11,0n-20#11)-> (0n-31l1112,0"-30i1112)... -> (1n-11i,1n ^) n—1 n—1

where clearly (0 L,0 0pis an edge in the binary hypercube [10]. Thus, one end (say the right end) of the

starting edge in the above sequence of edges traces a path living entirely in one sub-cube of dimension(n-l) whose

i bit is fixed at 0, namely

1 2 (n -1)

0n_10i ^0n_20i11 _30i1112... ^ 1n_10i

n—1 n—1 n

This path is from the NE 0 0. to a neighbor 1 0. of the NE 1 . Similarly, the left end traces a corresponding

path in the complementary sub-cube of dimension n-1 defined by i bit equal to 1, namely

1 2 (n -1)

0n_11i ^0"_21i11 ^ 1n_11i (5)

n—1 n n

Here we have a path from the neighbor 0 1. of the NE at 0 to the NE at 1 . Our approximation algorithm relies

on the well-known topological property of node disjoint path distribution in a binary hypercube.

Theorem 4.1 [10]: Let x and y be nodes in a binary hypercube of dimension k where the Hamming distance between x and y, H(x,y)=r for some l<r<k. Then (a) there are exactly r node disjoint paths each of length r between x and y (b) there are exactly (k-r) node disjoint paths each of length r+2 and (c) The set of all paths in groups a and b are node disjoint.

n—1 n—1

We apply this theorem to the two nodes x=0 1 .and y=1 1. with Hamming distance is n-1 in the sub-cube th

of dimension n-1 whose i bit is fixed at 1. Therefore, there are k=r=n-1 node disjoint paths each of length n-1 in that sub-cube.

Thus, if we denote the path in (5) succinctly as

0n_10i{1,2,...,n-l}ln_10i

where the {} term is the order of dimensions along which you move to the next node in the path from 0 0. to n-1

1 0.. It can be shown [10] that by left circular shift the elements in the ordered set we can generate the n-1 node disjoint paths in the n-1 dimensional sub-cube. These are given by

1 2 3 ... n -2 n -1

n -112

n -1 1 12

n - 3 n - 2

Likewise there is a corresponding sequence of paths in the other sub-cube given by

1 2 3 ... n -2 n -1

n -112

n -1 1 12

n - 3 n - 2

1 "1..

Thus, for each player i, there are n-1 distinct increasing sequences of inequalities that start from a negative number

and end in a positive number. Hence, for all n players, there are a total of n(n~\)=O(n ) sequences and hence O(n )

subsets of players who are potential candidates for the tipping set. Our goal is to find a minimal tipping set from 2

these O(n ) candidate subsets.

5 An Example

Consider an example of a n=4 player airline game with the following values of the parameters:

L. = 1000,: c. = 99,: p. = 0.1 for all l<i<4 and the matrix of q's given by

0.99.02.02 10 0 0 0.99 0 .02 L 0.99.02 0 .

Then using the expressions (2),(3), and (4) we can readily compute the 16*4 payoff (or loss in our case) table for

each of the 2 =16 plays for each of the 4 players. For lack of space, we will not explicitly show this table. It can be

4 4 4 4 4

verified that for this choice of parameter values, 0 and 1 are two NE with 1 being better than 0.1 is Pareto

optimal in this case. Applying the approximation algorithm developed in Section 4, we get three disjoint sequences

of inequalities induced by the three node disjoint paths for Player 1 as follows. Referring to the binary hypercube in Figure1a the first disjoint sequence is given by

u1(1000)-u1(0000)>u1(1100)-u1(0100)>u1(1110)-u1(0110)>u1(1111)-u1(0111)

which for the above example becomes 99>—l>—l>—l.

It follows that {2} is a candidate tipping set for player 1. That is, if player 2 changes his strategy from 0 to 1, some other players can reduce their losses by switching from 0 to 1. Thus, player 2 has influence over other those other players

Similarly, the second sequence

u1(1000)-u1(0000)>u1(1001)-u1(0001)>u1(1101)-u1(0101)>u1(1111)-u1(0111)

which for the above example becomes 99>99>-l>-l

from which we get {2,4} as a candidate tipping set. From the third sequence

u1(1000)_ u1(0000) u1(1010)_ u1(0010) u1(1011)_ ^(0011) u1(1111)_ u1(0111)

which leads to 99>99>99>-l.

from which we get {2,3,4} as a candidate tipping set.

By repeating the above procedure for Players 2, 3, and 4, we can obtain three candidate tipping sets for each of the players, which is summarized in Table 2

Table 2:

candidate tipping sets Path 1 Path 2 Path 3

T1 {2} {2,4} {2,3,4}

T2 {1,3,4} {1,3,4} {1,3,4}

T3 {1,2,4} {1,4} {1,2,4}

T4 {1,2,3} {1,3} {1,2,3}

Our goal is to find a minimal tipping set from these n(n-l)=12 candidate tipping sets.

In the following Section, we describe a simple algebraic method for extracting a minimal tipping set.

6 An Algebraic Method

Previously in [11] we transformed similar facts about state independent games into a graph form to examine the influence patterns. We constructed these influence graphs by representing each player by a node i with an edge from player 2 to player 1 if player 2's choice to invest impacts or influences player 1 directly.

Generating such a graph for this example gives us the graph in Figure 1b. General super modular games lack some of the structure we exploited in state independent games. Unlike state independent games, one node's influence is not enough to tip given node. In this example, all of the influencing nodes are needed to tip a node, thus adding complexity. To address this added challenge, we have developed an algebraic method for finding a minimal tipping set.

Let 2={0,1}, let + and * denote the usual logical AND and OR operators respectively. For simplicity, a*b is denoted by ab. Then, (£,+,*) defines a commutative semi-ring. For all a,b^L, we have a+a=a, aa=a, a+ab=a(1+b)=a.

Let a,b,c,d be the four binary variables corresponding to players 1 though 4 respectively. We now encode a subset {1,2,4} of players by abd. Accordingly, the information in Table 2 can be encoded as follows:

T1=b+bd+bcd=b T2=acd+acd+acd=acd

T3=abd+ad+abd=ad (6)

T4=abc+ac+abc=ac

These expressions represent the conditions for each player to choose to invest. T1 gives the conditions for player a, T2, for b, and so on.

First, we can algorithmically trim the number of players to examine. While this is not needed in order to get an answer, it improves both the accuracy of the approximation and the typical run time.

1) Any case in which a player's action is determined entirely by a single other player, we can collapse those players. In the context of the game, this is because the influences of the impacting player contain the influences of the impacted player. In this example, b completely determines the action of a. This means that the impact of b contains the entire impact of a, and that when picking minimal tipping sets, there is no reason to pick a. Collapsing a into b requires moving a's influences to b. This is to make sure that b now influences all players that a previously did. This should be done one at a time in order to stop cycles when a node is influenced only by itself. If no such players exist, we simply move on to the next step. Then from (6) we get:

T2=bcd;T3=bd;T4=bc

2) We also do not want to double count a player. If a player is externally motivated to invest, we do not care about the conditions to tip that player. To reflect this, we add to each expression the set consisting only of that player. This yields:

T2=b+bcd=b; T3=c+bd;T4=d+bc

Now we are ready for the core procedure. We want the value of each of these expressions to be 1, which is the same as their product being 1. Thus we want to find the minimal subset variables to set to 1 such that T2*T3*T4=1. Substituting from above we get:

(b)(c+bd)(d+bc)=(bc+bd)(d+bc)=(bcd+bc+bd+bcd)=1

Thus we have candidate tipping sets of bcd,:bc,:bd:bcd. Any of these will give a tipping set, but since we are interested in minimal tipping sets, we take one of the smaller sets bc or bd. to choose for our minimum set.

7 The Uniform Case

One special form of the Airline Security game is the uniform case. In this case, the agents behave identically with

pi=pV=p,ci=c/=c,Li=LV=L,qV=qk/=q

By examining the behavior of the approximations in this case along with the case's properties, we gain some insight regarding the performance of the method. Full proofs are omitted for space, but are available in the full report. The most relevant property of the uniform game is the following:

Lemma 7.1: A uniform game can be tipped by any set S of players with |^|>n-l- /gg(l-q) .

Corollary 7.1: In the uniform game, there can be no cascades.

Lemma 7.2: The hypercube left shift sampling will find a set of n-1 groups of |S| consecutive players to tip each individual player.

Theorem: Given the above a uniform game and n-1 groups of |S| players tipping each single player, the algebraic simplification will find the optimal solution. Example

Now we look at how the approximation system behaves in this context. Since any k players will tip any player. For players a,b,c, and d, set the following parameters:

j?=0.1,c=99,^=.009,L=1000

For clarity, examine what happens when |S|=2. This means that any 2 players will tip any player, thus each player's set of candidate tipping sets will be the first two corrected players and the corresponding first two from every left shift. So for players a,b,c,d, we would have:

a:bc,cd,bd; b:cd,ad,ac ; c:ab,bd,ad; d:ab,bc,ac Now, when we apply the algebraic method on this set, we get:

a=a+bc+cd+bd ; b=b+cd+ad+ac ; c=c+ab+bd+ad ; d=d+ab+bc+ac This system of equations simplify as:

a*b*c*d=1

(a+bc+cd+bd)(b+cd+ad+ac)(c+ab+bd+ad)(d+ab+bc+ac)=1 (ab+ad+ac+bc+cd+bd)(cd+bc+ac+ab+bd+ad)=1

ab+ad+ac+bc+cd+bd=1 This finds several size 2 tipping sets, which is the optimal.

8. Conclusions

The problem of finding the minimal tipping set in a general n-person super modular games that arise in the context of airline security with two NE consists of two steps. First, is to enumerate (n-1)! distinct paths for the edges in the complete binary lattice (hypercube) of dimension n. Second is to process the resulting (n-1)! candidate sets to obtain the minimal tipping set. Clearly, both the steps can take exponential time. DDIn this paper, we have given an algorithm to enumerate a set of O(n) node disjoint paths to obtain O(n) candidate sets by exploiting the topological properties of the underlying binary hypercube. Using a simple algebraic method, we then reduce these candidate sets into a single tipping set, which is an approximation of the minimal tipping set. The approximation is then shown to be optimal in the case of uniform airline security games. □□ Efforts to quantify the quality of this approximation of the tipping set are currently under way.

References

1. Owen, G. (1995) Game Theory, Academic Press (Third Edition), 447 pages.

2. Heal, G.M. and H. Kunreuther, (2005) "IDS Models of Airline Security",,Journal of Conflict Resolution, Vol 41, pp 201-217.D

3. Heal, G.M., and H. Kunreuther (2006), Supermodularity and Tipping, National Bureau of Economic Research, 1050 Mass Ave, Cambridge, MA.

4. Kunreuther, H. and G.M. Heal, (2003) "Interdependent Security," Journal of Risk and Uncertainty, Special Issue on Terrorist Risks, Vol 26, No 2/3, pp 231-249.

5. Nash, J. (1951) "Noncooperative games," Ann. Math. 54, 289-295.□

6. Gladwell, M (2000) Tipping Point, Little Brown and Co., New York.

7. Schelling, T. (1978) Micromotives and Macrobehavior, Norton, New York.

8. Dixit, A. K. (2002) "Clubs with Entrapment," American Economic Review, Vol 93, No. 5, pp 1824-1829.

9. Topkis, D. (1998) Supermodularity and Complementarity, Princeton University Press, Princeton, NJ.

10. Lakshmivarahan, S., and S. K. Dhall (1990) "Analysis and design of Parallel Algorithms", McGraw Hill, New York Chapter 2.

11. Cremeans, B., S. Lakshmivarahan, S. K. Dhall (2011) , "Impact of State Independence on Tipping in Interdependent Airline Games" Technical Report, School of Computer Science, University of Oklahoma, Norman, OK 73019, USA, September 2011 □□

12. Dhall, S. K. and S. Lakshmivarahan, and P. Verma (2009), On the Number and the Distribution of the Nash Equilibria in SuperModular Games and their Impact on the Tipping Set, Proceedings of the International Conference on Game Theory and Networks, Game Nets, 2009, Istanbul, pp 691-696.