Disease dynamics in a stochastic network game: a little empathy goes a long way in averting outbreaks

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1 Disease dyamics i a stochastic etwork game: a little empathy goes a log way i avertig outbreaks Supplemetary Material Ceyhu Eksi, Jeff S. Shamma ad Joshua S. Weitz A Existece of a MMPE strategy profile We recall the utility fuctio of idividual i defied i the Methods sectio below, u i a i,{a j } j Ni,st = a i β c 0 c s i t a j s j t c 2 s i t a j s j t. S Defie the space of probability distributios o the actio space 0 a i [0,] as [0,]. A mixed strategy profile σ is a fuctio that maps the state s {0,} to the space of probability distributios o the actios space, i.e., σ i : {0,} [0,]. The defiitio of a mixed MMPE strategy profile σ := {σi : {0,} [0,]} is a distributio o the actio space that satisfies the followig, for all t =, 2,..., ad i N, u i σ i st,σ N i st,st u i σ i st,σ N i st,st for ay σ i {0,} [0,] where σn i := {σ j : j N i }. From the existece of a mixed Nash equilibrium i games with cotiuous payoffs, we kow that a mixed MMPE exists for the game with payoffs S. I the followig we costructively show that i the stochastic disease etwork game with payoffs i S, a degeerate pure MMPE strategy profile σ := {σi : {0,} [0,]} that satisfies the above relatio i S2 exists. Note that a degeerate distributio puts weight oe o a sigle actio value, that is, the strategy profile correspods to a sigle actio profile for ay state. Sice i the stochastic game populatio respose is determied by the curret state of the disease oly, i.e., equilibrium is statioary, it suffices to show the existece of a pure Nash equilibrium strategy for the stage game with state s {0,}. A pure Nash equilibrium NE strategy profile σ : {0,} [0,] of the stage game with payoffs S ad state s satisfies u i σ i,σ N i,s u i σ i,σ N i,s for ay σ i {0,} [0,]. We defie the correspodig equilibrium actio profile as a := σ s for a give state s. For a give idividual i with state s i {0,}, eighbors state s Ni := {s j } j Ni, ad eighbor actio profile a Ni we have the best respose of idividual i as follows, BR i a Ni,s i,s Ni := argmaxu i a i,a Ni,s Ni = a i [0,] c 0 > c s i a j s j + c 2 s i a j s j where is the idicator fuctio. Sice the payoffs are liear i self-actios, the actios that maximize the payoffs are i the extremes a i = or a i = 0 depedig o the states ad actios of their eighbors. We ca equivaletly represet the NE defiitio i S3 as a fixed poit equatio by usig the best respose defiitio, S2 S3 S4 S5 a i = BR i a N i,s i,s Ni i N. S6 I the followig we defie the cocept of strictly domiated actio which will be a useful i fidig the Nash equilibrium. Defiitio Strictly domiated actio For a give state s {0,}, a actio a i [0,] is strictly domiated if ad oly if there exists a actio a i [0,] such that u i a i,a Ni,s > u i a i,a Ni,s a Ni S7

2 Figure S. The iterated elimiatio process i Defiitio 3 o a 5 idividual etwork. The payoff costats are such that 2c > c 0 > c ad c 0 > 2c 2. Idividuals ad 2 are sick s =, s 2 = ad the rest are healthy. The cotact etwork is show at k = 0. At step k =, idividuals i = {,2,4,5} elimiate all actios except by S9 give c 0 > c ad c 0 > 2c 2. Idividual 3 caot elimiate ay actio from the space. At step k = 2, idividual 3 elimiates all actios except 0 by S0. Sice all idividuals have sigleto o-domiated actio spaces, i.e., N \ 2 = /0, the process eds. Furthermore, the correspodig actio profile is a Nash equilibrium of the stage game i S by Lemma 2. If a actio a i is strictly domiated the there exists a more preferable actio a i for ay circumstace. It is clear that if a actio is strictly domiated the it caot be a ratioal actio from S6. I a game we ca iteratively remove the strictly domiated actios, this process is called the iterated elimiatio of strictly domiated strategies ad is defied below. Defiitio 2 Iterated elimiatio of strictly domiated actios Set the iitial set of actios A 0 i = [0,] for all i, ad for ay k N set A k i = {a i A k i ai is ot strictly domiated give ay a Ni A k N i } S8 We deote the set of actios of idividual i that survive the iterated elimiatio by A i := k=0 A k i. Whe A i has a sigle elemet, we say A i is a sigleto. We formally state a iterated elimiatio process for the game i S. Defiitio 3 A iterated elimiatio process for the epidemic game Begi with actio spaces A 0 i = [0,] for all i N. Fix s {0,}. Iterate k =,2,... [k = odd] Let A k i = for all i k where { k := i N \ k l= l : c 0 > 2 [k = eve] Let A k i = 0 for all i k where { k := i N \ k l= l : c 0 < c s i c s i \ l=eve l l=odd l s j + c 2 s i s j + c 2 s i \ l=eve l s j l=odd l s j } } S9 S0 If k = /0 the stop iteratio of k. Our ext result shows that the process described above elimiates all strictly domiated strategies i fiite time. Lemma Cosider the process described i Defiitio 3. There exists a iteratio step k such that {A k i } i N actios that are ot strictly domiated for the game with payoffs S ad state s {0,}. are the set of Proof: First cosider the odd iteratio steps, k = odd. If the coditio iside the bracket i S9 holds the for ay actio that idividual j N i \ l=eve l takes, a i = domiates ay other actio a i [0,] \ {} by Defiitio. To see this first 2/4

3 recall the best respose of idividual i S5. Note that by the eve steps, idividuals j l=eve l have a j = 0 as the oly ot domiated actio. Hece the best respose of idividual i ca be rewritte as c 0 > c s i \ l=eve l a j s j + c 2 s i \ l=eve l a j s j where we remove eighbors that oly have zero as the ot domiated actio. If the iequality iside the idicator fuctio is true eve whe the remaiig eighbors of i take actio, the a i = domiates all the other actios [0, of i, that is, for all a j [0,] for j N \ i, c 0 > c 0 > c s i c s i \ l=eve l s j + c 2 s i \ l=eve l \ l=eve l a j s j + c 2 s i s j \ l=eve l a j s j Assumig the left had side of the above iequality is oe the by the defiitio of strictly domiated actio i Defiitio, a i = is the oly actio that is ot domiated. Next, we cosider the eve iteratio steps, k = eve. If the coditio iside the bracket i S0 holds the for ay actio that idividual j N i \ l=odd l takes, a i = 0 domiates a i [0,] \ {0} by Defiitio. To see this first recall the best respose of idividual i S5. Note that by the odd steps, idividuals j l=odd l have a j = as the oly ot domiated actio. Hece the best respose of idividual i ca be rewritte as c 0 > + c 2 s i c s i \ l=odd l \ l=odd l a j s j + a j s j + l=odd l l=odd l s j S S2 s j S3 where we separate eighbors j N i l=odd l that oly have a j = as the ot domiated actio from the other eighbors j N i \ l=odd l. If the iequality iside the idicator fuctio is false eve whe the remaiig eighbors of i take actio 0, the a i = 0 domiates all the other actios 0,] of i, that is, for ay a j [0,] for j N i \ l=odd l c 0 < c s i s j l=odd l c 0 < c s i + c 2 s i \ l=odd l \ l=odd l + c 2 s i a j s j + a j s j + l=odd l l=odd l l=odd l s j s j s j S4 Assumig the left had side of the above iequality is oe the by the defiitio of strictly domiated actio i Defiitio, a i = 0 is the oly actio that is ot domiated. Further if o idividual ca elimiate a domiated strategy k = /0 at a iteratio k the at followig iteratios k +,k + 2,... there wo t be ay idividuals that elimiate ay actios as strictly domiated. To see this, assume the opposite is true, that is, k + /0. Sice k = /0 the the coditio i k + is idetical to the coditios iside S9 or S0 for k depedig o whether k + is odd or eve, respectively. There caot be a idividual that satisfies the coditios at iteratio k + because k + is selected amog idividuals that have ot bee previously i ay set, i.e., i N \ k l= l. This cotradicts k + /0. As a result, at each iteratio k the umber of idividuals i k has to be positive util a iteratio step k at which either there is o idividual left N \ k l= l = /0 or there is o idividual that satisfies the coditio after the colo i S9 or S0. Suppose ow that the process stops at iteratio k, i.e., k = /0, but there exists a idividual i N \ k l= l with strictly domiated actio S7 a i A k i. Suppose k is odd the by S2, ay actio a i A k i \ {} must be domiated by a i =. Suppose k is eve, the by S4 ay actio a i A k i \ {0} must be domiated by a i = 0. The k /0 which is a cotradictio. This 3/4

4 together with the fact that if k = /0 the l = /0 for l > k implies that if the process i Defiitio 3 stops all the strictly domiated strategies are elimiated. Fially, the fact that the umber of elemets of k is positive at every iteratio except the last iteratio i Defiitio 3 implies that the iteratio eds i at most iteratios. The followig Lemma shows that oce all the strictly domiated strategies are elimiated, the actio profile that assigs socialize to susceptible idividuals ad do ot socialize to ifected idividuals is a pure Nash equilibrium strategy profile. Lemma 2 Cosider the game defied by the payoffs i S give state s {0,} ad the iterated elimiatio process i Defiitio 3. Deote the actio space of i N that is ot strictly domiated by A i. Let a i = A i if A i is a sigleto. If A i is ot a sigleto the let a i = for s i = 0 ad a i = 0 for s i =. The resultig actio profile a is a pure Nash equilibrium of the game. Proof: We will use the followig equivalet defiitio i S6 of Nash equilibrium for the stage game i S, a i = c 0 > c s i a js j + c 2 s i a j s j. S5 Note that at the ed of the elimiatio process i Defiitio 3, the actio space of a idividual i is either a sigleto or is equal to A i = [0,]. A strictly domiated actio caot satisfy the above equatio, hece the equilibrium actio of a idividual with a sigleto o-domiated actio space is give by a i = A i. Now cosider i N such that A i = [0,]. Suppose a i = for a susceptible idividual s i = 0 is ot a Nash equilibrium actio. The idividual i ca deviate ad by the above equatio it must be that a i = 0 is a Nash equilibrium actio because c 0 < c a js j. S6 Note that by our assumptio a j = 0 for all the ifected idividuals s j = at which 0 is ot a strictly domiated actio, i.e., whe A j = [0,]. Hece i the right had side of the above iequality oly idividuals that do ot have 0 i their ot domiated actio set will matter. Furthermore, if 0 is a strictly domiated actio, socialize actio is the oly remaiig ot strictly domiated actio. Hece, we ca write the above iequality as follows, c 0 < c s j. :{0}/ A j S7 Now ote that if the above iequality is true the should be strictly domiated by actio 0 for idividual i. Hece, it is a cotradictio to the fact that the actio space is ot a sigleto, A i [0,]. A similar cotradictio argumet ca be made for the ifected idividuals s i = ad the equilibrium actio a i = 0. Therefore, the actio profile described i the statemet must be a pure Nash equilibrium of the game. Lemma shows that the process i Defiitio 3 elimiates all domiated actios i fiite time. Furthermore, if all idividuals are icluded i the process, i.e., if N = k= k the we ed up with a sigleto actio profile that is ot strictly domiated. This meas the game has a uique pure Nash equilibrium by defiitio of strictly domiated actio. If at the ed of the process i Defiitio 3, if all idividuals are ot icluded i the process, i.e., N \ k= k /0 the the set of ot strictly domiated actios is ot a sigleto. Lemma 2 proposes a pure strategy profile that is a Nash equilibrium of the game i S for the case that actio spaces of idividuals that survive strict elimiatio process are ot all sigleto. Lemmas ad 2 cosidered the stage game with payoffs S give state s {0,}. A MMPE strategy profile σ is a mappig from ay state to the actio space. We ca obtai a pure MMPE strategy profile usig Lemma 2 for all possible states s {0,}. That is, we use the elimiatio process i Defiitio 3 ad the actio assigmet give i Lemma 2 for all the states to costruct a MMPE strategy profile. I our simulatios, i this paper, we costruct the MMPE equilibrium strategy profiles followig this process see Fig. S for a example. B Price of Aarchy of the stage game I this sectio, we cosider the sub-optimality of the decisios of idividuals that play accordig to a MMPE strategy profile. By the defiitio of MMPE strategy, idividuals cosider curret payoffs ad play accordig Nash equilibrium strategy of that stage game. I the followig we show the worst stage game Nash equilibrium strategy ca be fold worse tha the optimal actio profile give a etwork ad state. 4/4

5 Defie the stage game Γs,G,c with payoffs i S parametrized by the disease state s, cotact etwork G ad utility costats c := {c 0,c,c 2 }. The welfare value of the actio profile a i this game is the sum of utilities of the idividuals, Wa,st = i= u i a i,a Ni,st = i= c 0 a i c + c 2 i, j E s i ts j ta i a j. S8 Note that the first summatio is over all the idividuals ad the secod summatio is over all the edges E i the etwork G. We defie the optimum actio profile as the maximizer of the welfare fuctio above, i.e., a opt = argmax a Wa,st. We deote the set of Nash equilibrium actio profiles by A ad defie the price of aarchy PoA as the ratio of the worst possible Nash actio profile to the optimum actio profile, PoA := mi a A Wa Wa opt. Obviously, PoA. I the followig we provide a lower boud o the price of aarchy. Propositio For a game Γs,G,c, the price of aarchy S9 has the followig lower boud PoA max i N N i maxc,c 2 c 0 where N i is the degree coectivity of i. Proof: Cosider the derivatives of u i i S ad W i S8 with respect to a i, respectively, u i = c 0 c s i s j a j + c 2 s i s j a j W = c 0 c + c 2 First ote that u i W W = u i s i s j a j + s i s j a j for ay s ad a i because c 2 s i s j a j + c s i s j a j S9 S20 S2. S22. S23 Therefore, give actios of others a i ad state s, it could be that u i > 0 while W < 0. Cosider the optimal ad equilibrium actio profiles where a opt j = a j for j N \ i, ad aopt i = 0 but a i =. That is, we have u j a j < 0 ad W a j < 0 for all j i but u i > 0 while W < 0. The, the differece betwee the equilibrium profile welfare Wa,s ad optimal actio profile welfare Wa opt,s is equal to S23, that is, Wa,s Wa opt,s = W We divide the differece above by Wa opt,s to get S24 + Wa W,s Wa opt,s = Wa opt,s. S25 Now assume u i = ε > 0 to get Wa ε c 2 s i,s j Ni s j a j + c s i j Ni s j a j Wa opt,s = + Wa opt. S26,s A trivial lower boud o the term for the umerator above is maxc,c 2 N i for ay state s. Furthermore a trivial upper boud for Wa opt,s is c 0. Usig these bouds we obtai a boud for the above equality, Wa,s Wa opt,s maxc,c 2 N i c 0 S27 5/4

6 Figure S2. Price of aarchy i the star etwork Example. We cosider the strog empathy c 0 < c 2 ad strog averseess c 0 < c case i Figure 2d of the mai mauscript. Left ad right figures show the two stage game pure Nash equilibria. The Nash equilibrium actio profile show by the right figure is optimal with welfare equal to c 0. The Nash equilibrium actio profile show by the left figure attais a welfare equal to c 0. The PoA is equal to /. Now ote that the choice of the idividual i is arbitrary. Whe this idividual is the idividual with the largest umber of eighbors we get a lower boud o the right had side of the above iequality which is the worst case lower boud o the price of aarchy i S20. Note that this upper boud ca be arbitrarily bad, i.e., i the order of /. I the discussio followig Figure 2d i the mai text we show that this boud is tight. We repeat this example below. Example Cosider a star etwork with idividuals. The ceter idividual with eighbors is the oly ifected idividual. Payoff costats are such that c 0 < c ad c 0 < c 2. There are two stage game Nash equilibria: a opt i = if s i = 0 ad a opt i = 0 if s i =, ad 2 a i = 0 if s i = 0 ad a i = if s i =. The first Nash equilibrium is also the optimal actio profile a opt yieldig a welfare of c 0. The secod Nash equilibrium a obtais a welfare of c 0 resultig i a PoA of Aother example o a star etwork follows. Example 2 Cosider the star etwork show i Figure S2. The ceter idividual has eighbors ad is the oly ifected idividual. Costats are such that c 0 = c + ρ ad c 0 = c 2 + ρ 2 for arbitrarily small positive costats ρ > 0 ad ρ 2 > 0, so that c 0 < c + c 2. The uique NE actio profile is that all idividuals are social, a i = for all i N. The welfare for the NE is Wa = c 0 c + c 2 = ρ + ρ 2. The optimal actio profile is that oly the susceptible odes are social a opt i = for i ad a opt = 0. The welfare for the a opt is Wa opt = c 0. The PoA ρ /c 0 for large. Here, we see that PoA is determied by the closeess of c to c 0. Note that i this example, equilibrium is uique ad, ulike Example, the optimal actio profile is ot a Nash equilibrium. Aother otio that eables us to gauge the optimality of equilibria is the Price of Stability PoS. PoS is the ratio of the best possible Nash actio profile to the optimum actio profile,. PoS := max a A Wa Wa opt S28 Note that PoS by defiitio of a opt. We remark that i Example 2, the equilibrium is uique, meaig PoS = PoA. Therefore, ot oly the PoA but also the PoS ca be arbitrarily bad. C R 0 boud We formally defie the reproductive ratio R 0 i the followig. 6/4

7 Defiitio 4 Let the iitial state of the populatio be give by s where s i =, otherwise s j = 0 for all j i for some radomly selected idividual i. The R 0 is the expected umber of idividuals that cotract the disease from the radomly selected idividual i util i heals, [ R 0 := E E [ t= j= s j t + s j t =,i js i l = for l < t s ]] where s j t + s j t =,i j is the idicator fuctio that is oe if idividual j trasitios to a ifected state at time t ad i is the oe ifectig j, ad s i l = for l < t is the idicator fuctio that is oe if idividual i has ot healed yet. The outside expectatio is with respect to the uiform distributio that selects the iitial ifected idividual, ad the iside expectatio is with respect to the trasitio probabilities of the Markov chai. I the followig, we derive bouds for R 0 give that idividuals act accordig to a MMPE strategy profile ad we select the iitial ifected idividual radomly from the etwork. Theorem Cosider a etwork with degree distributio Pk. Assume the ifected idividual is chose from the populatio uiformly at radom. Scale c 0 with β for coveiece, i.e., c 0 := βc 0, ad assume c 0 > c.assume c 0 > c. The R 0 defied i S29 has the followig upper boud, R 0 β δ K k= kpk where K := mi c 0 /c 2,. Proof: We start by movig the secod expectatio iside the sum i the defiitio of R 0 S29 to get the followig, R 0 = E [ t= j= E [ s j t + s j t =,i js i l = for l < t s ] ]. Now cosider the coditioal expectatio iside the summatio which we ca equivaletly represet as the followig coditioal probability E[s j t + s j t =,i js i l = for l < t s] S29 S30 S3 = P s j t + s j t =,i j,s i l = for l < t s. S32 Note that the above coditioal probability is the probability that j is ifected by i at time t ad i remaied ifected util time t give i is ifected at t = 0. Usig the chai rule ad law of total probability, we ca write the above coditioal probability as follows Ps j t + s j t =,i j,s i l = for l < t s = P s j t + s j t =,i j s i t =,s j t = 0,s Ps i t =,s j t = 0 s i l = for l < t,s + P s j t + s j t =,i j si t = 0,s j t = 0,s Ps i t = 0,s j t = 0 si l = for l < t,s + P s j t + s j t =,i j si t =,s j t =,s Ps i t =,s j t = si l = for l < t,s + P s j t + s j t =,i j s i t = 0,s j t =,s Ps i t = 0,s j t = s i l = for l < t,s P s i l = for l < t s Note that the first four lies o the right had side equal to the probability that i ifects j at time t give that i remaied ifected util t by law of total probability. The last lie is the probability that i remaied ifected give that i started ifected sice s i =. Observe that the probability that i ifects j is zero if idividual i is susceptible at time t or idividual j is ifected, i.e., P s j t + s j t =,i j s i t = 0 = 0 or P s j t + s j t =,i j s j t = = 0. Hece oly the first lie of the four lie expressio is ozero which simplifies the idetity above as follows Ps j t + s j t =,i j,s i l = for l < t s = P s j t + s j t =,i j si t =,s j t = 0,s Ps i t =,s j t = 0 si l = for l < t,s P s i l = for l < t s S33 S34 7/4

8 The probability of healig at each step is idepedet, hece the last coditioal probability is equal to δ t. The coditioal probability that i remais ifected ad j is ifected at time t give that i is ifected util time t is less tha δ by the argumet below, Ps i t =,s j t = 0 s i l = for l < t,s = Ps j t = 0 s i t =,s i l = for l < t,sps i t = s i l = for l < t,s S35 Ps i t = si l = for l < t,s = δ. The first equality above follows by chai rule. The iequality follows by the fact that the probability is less tha oe. The secod equality is true by the trasitio probability of idividual i from a ifected state. Substitutig these idetities iside the equatio S34 we obtai the followig S36 S37 Ps j t + s j t =,i j,s i l = for l < t s P s j t + s j t =,i j si t =,s j t = 0,s δ t. S38 Now cosider the coditioal probability o the right had side of S38, the probability that j is ifected at time t + by i give that j is susceptible ad i is ifected at time t. We have the followig upper boud, Ps j t + s j t =,i j si t =,s j t = 0,s = P s j t + =,i j si t =,s j t = 0,s = P s j t + =,i j a i t =,a jt =,s i t =,s j t = 0,s P a i t =,a jt = s i t =,s j t = 0,s P s j t + =,i j a i t =,a jt =,s i t =,s j t = 0,s P a i t = si t =,s j t = 0,s = β j N i P a i t = si t =,s j t = 0,s S39 The first equality is by the fact that s j t + = if s j t = 0 ad s j t + s j t =. The secod equality above is by the law of total probability ad by the fact that if i or j takes a actio to self-isolate, i.e., a j t = 0 or a i t = 0, the i caot ifect j. The iequality follows by the fact that P a i t =,a j t = si t =,s j t = 0 P a i t = si t =,s j t = 0. The last equality follows because if both agets socialize at ormal levels the the ifectio probability is β whe aget j ad i are coected, j N i. Now we cosider a i t. We kow that the MMPE actio of idividual i is a best respose to best respose actios of eighbors from S6 where best respose fuctio is give by S5. Specifically from the perspective of a ifected idividual i, the best respose actio is give by the followig a i t = c 0 > c 2 a jt s j t S40 Cosider t =. If c 0 > c, the a j = for all j N i. Hece, if c 0 > c 2 N i the a i =, i.e., a i = c 0 > c 2 N i. Also ote that if a i = the a i t = for all t >. Moreover, if a i = 0 the i ever ifects aother ode if it is the oly iitially ifected idividual. Hece, we have a i t = a i for all t. That is, the actio of aget i at time t is determied by the iitial actio ad is idepedet of the state at time t. Therefore, we ca write P a i t = si t =,s j t = 0,s = P a i = s = c 0 > c 2 N i. S4 Substitutig the above idetity i S39 ad usig S39 i S38, which the we substitute to S32, we get the followig upper boud for R 0 i S3, R 0 E [ t= j= βc 0 > c 2 N i j N i δ t ]. S42 8/4

9 Now ote the idicator fuctio terms deped o the etwork of idividual i but they do ot deped o the idetities of the eighbors. I fact, j= j N i is equal to the umber of eighbors of i, i.e., N i. I additio, we have t= δt = /δ. Usig these idetities, we ca write the above boud as follows, R 0 β δ E [ N i c 0 > c 2 N i ] S43 Idividual i is chose uiformly radom ad it has k eighbors with probability Pk, therefore, the expectatio o the right had side is equal to the followig, R 0 β δ k= kk < c 0 c 2 Pk S44 Boud i S30 follows. D R 0 boud scale-free Corollary Cosider a radom scale free etwork where the degree distributio follows Pk k γ for γ = 2. The R 0 defied i S29 has the followig upper boud, R 0 β 2 δ logmi c 0/c 2, +. S45 Proof: We directly use the boud i S30 ad substitute i Pk = L 2,k 2 where L 2, = k= k 2 is the ormalizatio costat for the scale-free distributio R 0 L 2, β δ mi c 0 /c 2, k k= S46 We first upper boud the ormalizatio costat L 2, by the fact that k= k 2 > 2. Hece, L 2, 2. Next we ote that the summatio behaves logarithmically which yields the desired boud i S45. E R boud We formally defie the reproductive ratio R i the followig. Defiitio 5 Let the iitial state of the populatio be give by s where s i =, otherwise s j = 0 for all j i for some kpk k kpk. radomly selected idividual i. The probability that we select a iitial sick idividual with degree k is give by Qk := The R is the expected umber of idividuals that cotract the disease from a radomly selected idividual i util i heals, [ R := E E s j t + s j t =,i js i l = for l < t ]] s S47 [ t= j= where s j t + s j t =,i j is the idicator fuctio that is oe if idividual j trasitios to a ifected state at time t ad i is the oe ifectig j, ad s i l = for l < t is the idicator fuctio that is oe if idividual i has ot healed yet. The outside expectatio is with respect to the probability distributio Qk, ad the iside expectatio is with respect to the trasitio probabilities of the Markov chai. Below we preset a boud for R for a geeric etwork. Theorem 2 Cosider a etwork with degree coectivity distributio Pk. Assume c 0 > c. The R defied i Defiitio 5 has the followig upper boud, R β δ mi c 0 /c 2, k= k 2 Pk k= kpk S48 9/4

10 β = 0. β = 0.2 β = 0.3 Figure S3. Effect of risk averseess c ad empathy c 2 costats o mea eradicatio time. We fix the healig rate to δ = 0.2 ad the populatio size to = 00. The ifectio rate β values equal to 0.,0.2, ad 0.3 for figures left, middle, ad right, respectively. We let c 0 = for each figure. The axes i the figures correspod to the costat values of c ad c 2. For a give value of c ad c 2, we geerate 50 scale-free etworks usig the preferetial attachmet algorithm ad ru the stochastic disease etwork game for 200 steps for each etwork. Each ru starts with all idividuals ifected. The grid color represets the average eradicatio time amog rus where we let eradicatio time be equal to 200 if the disease is ot eradicated. The eradicatio time decreases as c or c 2 icreases i the regio where disease is eradicated fast, i.e., R >. Proof: First ote that the oly differece betwee the R 0 defiitio ad R is the differece i probability distributios of selectig the iitial ifected idividual. Hece, the boud i S43 applies to R which we repeat here for coveiece, R β δ E [ N i c 0 > c 2 N i ] S49 where ow the expectatio is with respect to the distributio Qk. Therefore, the expectatio o the right had side is equal to the followig, R β δ k= kk < c 0 c 2 Qk S50 Boud i S48 follows by usig the defiitio of Qk = F R boud scale-free kpk k kpk. Corollary 2 Cosider a radom scale free etwork where the degree distributio follows Pk k γ for γ = 2. The R give by Defiitio 5 has the followig upper boud, R β δ mi c 0 /c 2,. log S5 Proof: We directly use the boud i S48 ad substitute i Pk = L 2,k 2 where L 2, = k= k 2 is the ormalizatio costat for the scale-free distributio R β δ mi c 0 /c 2, k= L 2, L 2, l= l = β δ mi c 0 /c 2, k= l= l. S52 The boud i S5 follows by otig that l= l log. G Mea eradicatio time See Figure S3 for the effect of parameters c, c 2, β o eradicatio time. H Average ifectivity levels See Figure S4 for the effect of parameters c, c 2, β o ifectivity level. 0/4

11 β = 0. β = 0.2 β = 0.3 Figure S4. Effect of risk averseess c ad empathy c 2 costats o average ifectivity level. We fix the healig rate to δ = 0.2 ad the populatio size to = 00. The ifectio rate β values equal to 0.,0.2, ad 0.3 for figures left, middle, ad right, respectively. We let c 0 = for each figure. The axes i the figures correspod to the costat values of c ad c 2. For a give value of c ad c 2, we geerate 50 scale-free etworks usig the preferetial attachmet algorithm ad ru the stochastic disease etwork game for 200 steps for each etwork. Each ru starts with all idividuals ifected. The grid color represets the average of ifectivity level at time t = 200 amog rus. Icreasig c reduces the average umber of ifected i the populatio whether the disease is edemic or ot. I Critical empathy thresholds for modified scale-free etworks I the preferetial attachmet model, used i Figures 4-6 of the mai text, a ode is added at each step ad is coected to a existig ode selected radomly proportioal to their degree see the Barabasi-Albert algorithm. A modificatio to this algorithm is whe each added ode is coected to two existig odes selected radomly proportioal to their degree. This modified algorithm geerates scale-free etworks with a expected scalig degree of γ = 2.3 while the o-modified preferetial attachmet algorithm geerates a expected scalig degree of γ = 2. I Figure S5, we test the accuracy of the bouds for R 0 ad R i equatios S30 ad S48 equatios ad 4 i the mai text for scale-free etworks geerated accordig to the modified preferetial attachmet algorithm. Usig these iequalities ad substitutig for Pk = L 2.3,k 2.3 where L 2.3, is a ormalizig costat, we solve for empathy costat values c 2 that make the right had side of equatios ad 4 less tha oe see the captio of Fig. S5 for the critical empathy values. We ote that the closed form critical thresholds of the empathy costat c 2 i equatios S45 ad S5 equatios 3 ad 6 of mai text are obtaied whe the etwork is scale-free with scalig degree γ = 2. The critical empathy values obtaied for γ = 2.3 are greater tha the critical empathy values whe γ = 2. Numerical experimets show that the critical empathy costats that make R < are still good idicators of fast disease eradicatio. J Sesitivity to iitial level of ifectivity We assess the robustess of the critical empathy threshold to iitial coditios by cosiderig iitial ifectivity levels of {5%,20%,50%} i Figure S6. Besides the differece i iitial coditios, the umerical setup is idetical to Figure 6 i mai text. The umerical experimets cofirm that the critical empathy threshold for R i S5 is a good idicator for disease eradicatio. There are oly a few rus which fail to eradicate the disease withi 200 time steps whe the critical empathy for R < is exceeded refer to the captio of Figure S6 for details. K Epidemic threshold whe c 2 = 0 If c 2 = 0, a MMPE actio profile at time t ca simply be writte as follows, a i t = c 0 > c s i t s j t. S53 That is, the ifected will socialize ormally at all times ad a susceptible aget will socialize if the umber ifected aroud is less tha c 0 c. Give the MMPE profile above, the ifectio probability of a susceptible idividual i is give by p i 0 t = Ps it + = s i t = 0 = βa i ts j t. S54 /4

12 Figure S5. Effect of risk averseess c ad empathy c 2 costats o disease eradicatio with scale-free etworks with higher scalig degree. We cosider = 00 idividuals, ad let δ = 0.2, ad c 0 =. The scale-free etwork is geerated accordig to the modified preferetial attachmet algorithm resultig i a scale-free etwork with average scalig degree of γ = 2.3. The average ode degree is The critical values of c 2 {0.04,0.35,.0} that make R 0 < for β = {0.,0.2,0.3} calculated usig equatio S30 are marked with white dotted dashed lies. The critical values of the empathy costat, c 2 = {0.2,0.27,0.35} marked with red solid lies, that make R < for β = {0.,0.2,0.3} calculated usig equatio S48 are still accurate i determiig fast eradicatio. 2/4

13 Figure S6. Effect of risk averseess c ad empathy c 2 costats o disease eradicatio whe iitial umber of ifected is 5% top, 20% middle, ad 50% bottom of the populatio. We cosider = 00 idividuals, ad let δ = 0.2, ad c 0 =. For a give value of c ad c 2, we geerate 50 scale-free etworks usig the preferetial attachmet algorithm ad ru the stochastic disease etwork game for 200 steps for each etwork. The grid color represets the ratio of rus i which disease is eradicated withi 200 steps. For figures left, middle, ad right the eradicatio of threshold βλ max A/δ is equal to 2.65, 5.3, ad 8, respectively. The critical values of c 2 {0.02,0.6,0.36} that make R 0 < for β = {0.,0.2,0.3} calculated usig equatio S45 are marked with white dotted dashed lies. The critical values of the empathy costat, c 2 = {0.,0.22,0.33}, that make R < for β = {0.,0.2,0.3} calculated usig equatio S5 are accurate i determiig fast eradicatio for ay value of c marked with red solid lies. The rate of disease eradicatio is above 95% amog 50 trials whe the value of the empathy costat is above the threshold that makes R <. That is, for c 2 that make R < there are oly a few trials that fail to eradicate the disease withi 200 steps which are observed whe β = 0. ad 20% iitially ifected, ad whe β = {0.2,0.3}. 3/4

14 We ca boud the above coditioal probability as follows by usig the relatio j Ni βa i a j s j t βa i j Ni a j s j t, p i 0 t βa i t s j t. S55 Next we obtai a -state approximatio of the Markov chai dyamics see for a similar approximatio 2. Defie the probability of ifectio of idividual i at time t p i t = E[s i t]. Let pt [0,] be ifectio probability vector of the populatio. By law of total probability, p i t + = p i t δ + p i tp i 0 t. S56 Usig the upper boud above for p i 0 t ad otig that a i = c 0 > βc j Ni s j t from S53 if s i t = 0, we obtai p i t + p i t δ + p i tβ c 0 > c s j t s j t. S57 We ow approximate the above boud by replacig s j t terms with p j t to get p i t + p i t δ + p i tβ c 0 > c p j t p j t. S58 Whe we liearize the dyamics of pt defied by S58 aroud the fixed poit origi 0, i.e., the state of disease eradicatio p i t = 0 for all i, we get the followig, pt + = δi + βapt S59 where I is the idetity matrix ad A [0,] is the adjacecy matrix of the cotact etwork, i.e., its i jth elemet A i j = if j N i, otherwise A i j = 0. The approximate disease dyamics i S59 is liear. Furthermore, the origi 0 is a globally stable fixed poit if ad oly if λ max δi + βa < 3. We ca equivaletly write this coditio as β δ λ maxa <. This result implies that the state of disease eradicatio is ustable if β δ λ maxa >. Note that this threshold does ot deped o the risk averseess costat c. This implies that risk aversio caot by stop a outbreak whe c 2 = 0, except i the extreme case of c 0 < c. Refereces. Barabási, A.-L. & Albert, R. Emergece of scalig i radom etworks. Sciece 286, Paarpor, K., Eksi, C., Weitz, J. S. & Shamma, J. S. Epidemic spread over etworks with aget awareess ad social distacig. I Proceedigs of the 53rd Aual Allerto Coferece o Commuicatios, Cotrol, ad Computig, 5 57 Allerto, Illiois, USA, Va Mieghem, P., Omic, J. & Kooij, R. Virus spread i etworks. IEEE/ACM Trasactios o Networkig 7, DOI 0.09/TNET /4