A Framework for Optmal Investment Strateges for Competng Camps n a Socal Network PGMO Days 217 Swapnl Dhamal Postdoctoral Researcher INRIA Sopha Antpols, France Jont work wth Wald Ben-Ameur, Tjan Chahed, Etan Altman 14 November, 217 Swapnl Dhamal (INRIA) Optmal Investment Strateges for Competng Camps n a Socal Network 14 November, 217 / 19
Introducton Paper on arxv - Good versus Evl: A Framework for Optmal Investment Strateges for Competng Camps n a Socal Network Swapnl Dhamal (INRIA) Optmal Investment Strateges for Competng Camps n a Socal Network 14 November, 217 1 / 19
Fredkn Johnsen Model of Opnon Dynamcs v w w j w l j l w v w + w v + w j v j bas self j network v Swapnl Dhamal (INRIA) Optmal Investment Strateges for Competng Camps n a Socal Network 14 November, 217 2 / 19
Some Examples 1 Unbounded opnon values (v R) Fund collecton Sensors 2 Bounded opnon values (v [ 1, +1]) Electons Product adopton Swapnl Dhamal (INRIA) Optmal Investment Strateges for Competng Camps n a Socal Network 14 November, 217 3 / 19
Model of Opnon Dynamcs v g w g w w j j b w b w l l w v w bas v + w v self + w j v j j network + w g x good w b y bad x = nvestment by good camp on y = nvestment by bad camp on Swapnl Dhamal (INRIA) Optmal Investment Strateges for Competng Camps n a Socal Network 14 November, 217 4 / 19
Model of Opnon Dynamcs g b w g w b v w w j w l j l max x mn y v s.t. x k g, y k b x, y 1 (f bounded) w v w bas v + w v self + w j v j j network + w g x good w b y bad x = nvestment by good camp on y = nvestment by bad camp on Swapnl Dhamal (INRIA) Optmal Investment Strateges for Competng Camps n a Socal Network 14 November, 217 4 / 19
Convergence of Opnon Values and Computng v v τ = w v + w v τ 1 + w j v τ 1 j + w g x w b y j v τ = wv τ 1 + w v + w g x w b y v τ = lm η wη v τ η + ( w η) (w v + w g x w b y) η= v = (I w) 1 (w v + w g x w b y) Swapnl Dhamal (INRIA) Optmal Investment Strateges for Competng Camps n a Socal Network 14 November, 217 5 / 19
Convergence of Opnon Values and Computng v v τ = w v + w v τ 1 + w j v τ 1 j + w g x w b y j v τ = wv τ 1 + w v + w g x w b y v τ = lm η wη v τ η + ( w η) (w v + w g x w b y) η= v = (I w) 1 (w v + w g x w b y) 1 T v = 1 T (I w) 1 (w v + w g x w b y) r = ( I w T ) 1 1 Swapnl Dhamal (INRIA) Optmal Investment Strateges for Competng Camps n a Socal Network 14 November, 217 5 / 19
Convergence of Opnon Values and Computng v v τ = w v + w v τ 1 + w j v τ 1 j + w g x w b y j v τ = wv τ 1 + w v + w g x w b y v τ = lm η wη v τ η + ( w η) (w v + w g x w b y) η= v = (I w) 1 (w v + w g x w b y) 1 T v = 1 T (I w) 1 (w v + w g x w b y) r = ( I w T ) 1 1 v = r w v + r w g x r w b y Katz centralty of node s the th component of ( I αa T ) 1 1 Swapnl Dhamal (INRIA) Optmal Investment Strateges for Competng Camps n a Socal Network 14 November, 217 5 / 19
So we have Swapnl Dhamal (INRIA) Optmal Investment Strateges for Competng Camps n a Socal Network 14 November, 217 6 / 19 Result for Weghted Cascade-lke Models Proposton Let N = {j : w j }, d = N, and j N N j. If, w g = w b = w = 1 α+d = w j, j N, where α >, then r w g = r w b = 1 α,. ( I w T ) r = 1 r = 1 + w T r r = 1 + j N w j r j = 1 + j N ( 1 α + d j ) r j Let us assume that r = γ(α + d ), where γ s some constant. γ(α + d ) = 1 + j N γ = 1 + γd γ = 1 α
Model of Opnon Dynamcs (Concave Influence Functon) g b w g w b v w w w j w l j l max x mn y v s.t. x k g, y k b x, y 1 (f bounded) v w bas v + w v self + w j v j j network + w g x 1/p good w b y 1/p bad x = nvestment by good camp on y = nvestment by bad camp on Swapnl Dhamal (INRIA) Optmal Investment Strateges for Competng Camps n a Socal Network 14 November, 217 7 / 19
Investment Strateges (Concave Influence Functon) p x (r w g ) p p 1 p x (r w g ) p p 1 Swapnl Dhamal (INRIA) Optmal Investment Strateges for Competng Camps n a Socal Network 14 November, 217 8 / 19
Bounded Investment Per Node (Concave) ( : x, y 1) Proposton Let ˆγ > be the soluton of ( r w g tγ :r w g (,tγ] x =, f r w g x = 1, f r w g > tˆγ x = k g 1 :r w g >tˆγ ) t t 1 + :r w g >tγ (r w g ) t t 1 1 = k g :r w g (,tˆγ] (r w g ) t t 1, f r w g (, tˆγ] If there does not exst a ˆγ >, nvest 1 on all nodes wth r w g > and on all other nodes. The optmal strategy of the bad camp s analogous. Swapnl Dhamal (INRIA) Optmal Investment Strateges for Competng Camps n a Socal Network 14 November, 217 9 / 19
Bounded Investment Per Node (Concave) ( : x, y 1) Tral-and-error Iteratve Process Untl x 1, 1 Use the optmal strategy for the unbounded case 2 If for any, we get x > 1, assgn x = 1 to node wth the hghest value of r w g 3 Exclude node and decrement the avalable budget by 1 Swapnl Dhamal (INRIA) Optmal Investment Strateges for Competng Camps n a Socal Network 14 November, 217 1 / 19
Decson Under Uncertanty The good camp plays frst wth uncertan nformaton regardng w g, w b, w, whle the bad camp plays second max x k g x mn Eu f mn y k b y r w g x + r w v r w b y Swapnl Dhamal (INRIA) Optmal Investment Strateges for Competng Camps n a Socal Network 14 November, 217 11 / 19
Common Coupled Constrants ( : x + y 1) max ĵ max x x (r w g + max{r w b rĵwĵb, }) max{r w b rĵwĵb, } rĵwĵb k b Good camp chooses nodes wth not only good values of r w g, but also good values of r w b Node ĵ can be vewed as the node beyond whch the bad camp does not nvest on, as per ts preference orderng If the good camp does not nvest on node (that s preferred by bad camp over ĵ), the bad camp would beneft r w b rĵwĵb Swapnl Dhamal (INRIA) Optmal Investment Strateges for Competng Camps n a Socal Network 14 November, 217 12 / 19
Maxmn versus Mnmax ( : x + y 1) max x 1 max x 1 mn y 1 max x 1 mn y 1 x mn y 1 max x 1 mn y 1 x v max x 1 v = mn y 1 v mn y 1 v mn y 1 mn y 1 max x 1 max x 1 y max x 1 y v v v v Swapnl Dhamal (INRIA) Optmal Investment Strateges for Competng Camps n a Socal Network 14 November, 217 13 / 19
Common Coupled Constrants ( : x + y 1) max x mn y v mn y max x v Swapnl Dhamal (INRIA) Optmal Investment Strateges for Competng Camps n a Socal Network 14 November, 217 14 / 19
Two Phase Opnon Dynamcs s.t. max mn max mn x (1) y (1) x (2) ( ) x (1) + x (2) k g, y (2) v (2) ( ) y (1) + y (2) k b Swapnl Dhamal (INRIA) Optmal Investment Strateges for Competng Camps n a Socal Network 14 November, 217 15 / 19
Two Phase Opnon Dynamcs s.t. max mn max mn x (1) y (1) x (2) ( ) x (1) + x (2) k g, y (2) v (2) ( ) y (1) + y (2) k b Let = (I w) 1, then r = j j and s = j r jw jj j v (2) = s w v + ( ) s w g x (1) + r w g x (2) ( ) s w b y (1) + r w b y (2) Swapnl Dhamal (INRIA) Optmal Investment Strateges for Competng Camps n a Socal Network 14 November, 217 15 / 19
Two Phase Opnon Dynamcs s.t. max mn max mn x (1) y (1) x (2) ( ) x (1) + x (2) k g, y (2) v (2) ( ) y (1) + y (2) k b Let = (I w) 1, then r = j j and s = j r jw jj j v (2) = s w v + Loss to the bad camp f t acts myopcally ( ) s w g x (1) + r w g x (2) ( ) s w b y (1) + r w b y (2) k b (max (max {s w b, r w b, }) max { sîwîb, }) where î = arg max r w b Swapnl Dhamal (INRIA) Optmal Investment Strateges for Competng Camps n a Socal Network 14 November, 217 15 / 19
Two Phase Verson of Katz Centralty r = j j s = j r jw jj j w jj j j j j r j w ll l l l l r l m m m m r m r w mm s Swapnl Dhamal (INRIA) Optmal Investment Strateges for Competng Camps n a Socal Network 14 November, 217 16 / 19
Dependency between Parameters (One Camp, Unbounded) Let w g + w b be a constant θ ( 1 + w w g = θ v ) 2 ( 1 w w b = θ v ) 2 Swapnl Dhamal (INRIA) Optmal Investment Strateges for Competng Camps n a Socal Network 14 November, 217 17 / 19
Dependency between Parameters (One Camp, Unbounded) Let w g + w b be a constant θ ( 1 + w w g = θ v ) 2 ( 1 w w b = θ v ) 2 Clam It s optmal to ether exhaust the entre budget (k g (1) + k g (2) = k g ) or not nvest at all (k g (1) = k g (2) = ) It s an optmal strategy to nvest on at most one node n a gven phase Swapnl Dhamal (INRIA) Optmal Investment Strateges for Competng Camps n a Socal Network 14 November, 217 17 / 19
Opnon Dynamcs n Two Phases Let b j = r j wjj j and c = w v For each canddate optmal par (, j ) ncludng (, ) If θ θ j b j (c + 1) <, { = mn max k (1) g { kg 2 + s θ j b j If θ θ j b j (c + 1) >, then k(1) g = or k g b } } j c + r j θ b j (c + 1),, k g Swapnl Dhamal (INRIA) Optmal Investment Strateges for Competng Camps n a Socal Network 14 November, 217 18 / 19
Opnon Dynamcs n Two Phases Let b j = r j wjj j and c = w v For each canddate optmal par (, j ) ncludng (, ) If θ θ j b j (c + 1) <, { = mn max k (1) g { kg 2 + s θ j b j If θ θ j b j (c + 1) >, then k(1) g = or k g b } } j c + r j θ b j (c + 1),, k g Budget allotted for the frst phase 1 8 6 4 2 k g (1) for dependency case.2.4.6.8.95 Value of w Swapnl Dhamal (INRIA) Optmal Investment Strateges for Competng Camps n a Socal Network 14 November, 217 18 / 19
The Case of Competng Camps Under practcally reasonable assumptons w j, (, j) w, θ, v [ 1, 1], transform the problem nto a two-player zero-sum game wth each player havng (n 2 + 1) pure strateges show how the players utltes can be computed for each strategy profle show exstence of Nash equlbra and that they can be found effcently usng lnear programmng Swapnl Dhamal (INRIA) Optmal Investment Strateges for Competng Camps n a Socal Network 14 November, 217 19 / 19
Thank you! Paper on arxv - Good versus Evl: A Framework for Optmal Investment Strateges for Competng Camps n a Socal Network Swapnl Dhamal (INRIA) Optmal Investment Strateges for Competng Camps n a Socal Network 14 November, 217 19 / 19