The Impact of Tribalism on Social Welfare Matvey Soloviev (Cornell - PowerPoint PPT Presentation

The Impact of Tribalism on Social Welfare Matvey Soloviev (Cornell University) SAGT 2019, Athens, Greece joint work with Yuwen Wang and Seunghee Han 1

social welfare) in atomic linear routing, the PoA gets worse! Price of Anarchy is not due to selfishness Naive view of PoA Players are selfish and short-sighted, get stuck in bad equilibrium. Caragiannis (2010): If players are altruistic (act to maximise (2 5 3) 2

Price of Anarchy is not due to selfishness Naive view of PoA Players are selfish and short-sighted, get stuck in bad equilibrium. Caragiannis (2010): If players are altruistic (act to maximise 2 social welfare) in atomic linear routing, the PoA gets worse! (2 . 5 → 3)

Price of Anarchy is not due to selfishness Naive view of PoA Players are selfish and short-sighted, get stuck in bad equilibrium. Caragiannis (2010): If players are altruistic (act to maximise 2 social welfare) in atomic linear routing, the PoA gets worse! (2 . 5 → 3)

Price of Anarchy is not due to selfishness Naive view of PoA Players are selfish and short-sighted equilibrium. Caragiannis (2010): If players are altruistic (act to maximise 2 ? , get stuck in bad social welfare) in atomic linear routing, the PoA gets worse! (2 . 5 → 3)

Tribalism (1) In the real world, actual altruism may be a tall order. Democrats want to maximise the sum utility of Democrats, Republicans want to maximise the sum utility of Republicans... Uber cars want to maximise the sum utility of Uber cars... ants from anthill A want to maximise the sum utility of ants from anthill A... 3 At best, players care about their tribe:

Social context Other extreme: Social context games (player i weighs player j ’s Our model is “in the middle”. 4 utility by arbitrary factor p ij ). (Bilò et al. (2013) . . . )

Tribalism (2) Think of tribalism as players playing a game with different payoffs: Definition modified utility functions 5 G : game with utility functions u i . τ : a function that assigns players to tribes. The τ -tribal extension of G is the game G τ with the same players and strategies, and ∑ u τ i ( ⃗ u j ( ⃗ s ) = s ) . j ∈ N : τ ( i )= τ ( j )

Tribalism (3) The modified game can have different equilibria. 6 Definition However, we still rate them in terms of the original game: The Price of Tribalism (PoT) of a class of games G and partition functions T is i u i ( ⃗ sup ⃗ ∑ s ) s ∈ Σ 1 ×···× Σ n PoT ( T , G ) = sup , i u i ( ⃗ inf ⃗ ∑ s ) G ∈G ,τ ∈T G s ∈ S G τ where S G τ is the set of pure Nash equilibria of G τ .

Our results Game 4 3 Atomic linear routing 4 2 2 (convex rewards) Network contribution 2 1 1 (additive rewards) Network contribution k k ( k cliques) Social grouping 3 2 2 (2 cliques) Social grouping PoT Altruistic PoA PoA 7 2 k − 1 5 / 2

Social grouping (1) clubs, say If two players i and j are in the same club, they can be friends befriend each other. 8 Players i ∈ N want to socialise by joining one of two social Σ i = { A , B } . and get utility u ij ≥ 0. Players who are in different clubs don’t

Social grouping (2) B Here, every player gets a utility of 1. 1 1 1 1 Clearly, it’s optimal for everyone to be in the same club and 9 A d c b a But what if players start out in different clubs? befriend each other. 1 − ε 1 − ε 1 − ε 1 − ε This lower bound is tight: PoA = 2. (Also works for altruism.)

Friendship in the Time of Tribalism 1 friends in the same tribe count for twice as much! So nobody The friends in the other tribe would be twice as valuable, but 1 1 1 With tribes, however, the following is a Nash equilibrium: 10 B A d c b a 2 − ε 2 − ε 2 − ε 2 − ε wants to switch. This gives PoT ≥ 3.

Network Contribution Games (1) their relations. (Money, time...) function that tells us how much they’d gain from investing in their relationship, depending on each of their investments. think of this as a graph. 11 Each player i ∈ P has a budget B i they can divide up among For each pair of players e = { i , j } ∈ P ( 2 ) , have some symmetric Taking the pairs e with f e ( a , b ) = 0 to be non-edges, we can

f e a b f e a b f e a b Network Contribution Games (2) e.g. a b (“buying dinner”) a b (“hanging out”) a b (“synergy”) Get different PoA depending on what sorts of functions we allow. 12 The functions f e can take all sorts of forms.

f e a b f e a b Network Contribution Games (2) e.g. (“buying dinner”) a b (“hanging out”) a b (“synergy”) Get different PoA depending on what sorts of functions we allow. 12 The functions f e can take all sorts of forms. f e ( a , b ) = a + b

f e a b Network Contribution Games (2) e.g. (“buying dinner”) (“hanging out”) a b (“synergy”) Get different PoA depending on what sorts of functions we allow. 12 The functions f e can take all sorts of forms. f e ( a , b ) = a + b f e ( a , b ) = min { a , b }

Network Contribution Games (2) e.g. (“buying dinner”) (“hanging out”) (“synergy”) Get different PoA depending on what sorts of functions we allow. 12 The functions f e can take all sorts of forms. f e ( a , b ) = a + b f e ( a , b ) = min { a , b } f e ( a , b ) = a · b

Network Contribution Games (2) e.g. (“buying dinner”) (“hanging out”) (“synergy”) Get different PoA depending on what sorts of functions we allow. 12 The functions f e can take all sorts of forms. f e ( a , b ) = a + b f e ( a , b ) = min { a , b } f e ( a , b ) = a · b

Network Contribution Games (additive rewards) PoA (selfish and altruistic) is 1: everyone would always switch to the edge with highest c e , and this is in fact optimal. Not so under tribalism! 0 1 0 2 x y x y PoT 2 ( , actually) 13 Simple case: all functions are of the form f e ( x , y ) = c e ( x + y ) .

Network Contribution Games (additive rewards) PoA (selfish and altruistic) is 1: everyone would always switch to the edge with highest c e , and this is in fact optimal. Not so under tribalism! 0 1 0 PoT 2 ( , actually) 13 Simple case: all functions are of the form f e ( x , y ) = c e ( x + y ) . 2 ( x + y ) ( x + y )

Network Contribution Games (additive rewards) PoA (selfish and altruistic) is 1: everyone would always switch to the edge with highest c e , and this is in fact optimal. Not so under tribalism! 0 1 0 13 Simple case: all functions are of the form f e ( x , y ) = c e ( x + y ) . 2 ( x + y ) ( x + y ) ⇒ PoT ≥ 2 ( = , actually)

Network Contribution Games (coordinate convex rewards) (max, product...) unilateral!) Same under full altruism. What about tribes? 14 Less simple case: functions f e are convex in each coordinate. AH ’12: PoA = 2 under bilateral deviations. (Unbounded for

Network Contribution Games (coordinate convex rewards) 1 Stable against bilateral and even whole-tribe deviations! f 1 1 Nash 4 f 1 1 OPT f 15 f 1 1 1 1 1 ( 1 2 + ε ) f ε f ε f f ( x , y ) = x · y (anything with f ( x , 0 ) = 0 works)

Network Contribution Games (coordinate convex rewards) f Stable against bilateral and even whole-tribe deviations! f 1 1 Nash f 1 15 1 1 1 1 1 ( 1 2 + ε ) f ε f ε f f ( x , y ) = x · y (anything with f ( x , 0 ) = 0 works) OPT = 4 f ( 1 , 1 ) .

Network Contribution Games (coordinate convex rewards) f Stable against bilateral and even whole-tribe deviations! f 1 15 1 1 1 1 1 ( 1 2 + ε ) f ε f ε f f ( x , y ) = x · y (anything with f ( x , 0 ) = 0 works) OPT = 4 f ( 1 , 1 ) . Nash ≈ f ( 1 , 1 ) .

Network Contribution Games (coordinate convex rewards) f Stable against bilateral and even whole-tribe deviations! f 1 15 1 1 1 1 1 ( 1 2 + ε ) f ε f ε f f ( x , y ) = x · y (anything with f ( x , 0 ) = 0 works) OPT = 4 f ( 1 , 1 ) . Nash ≈ f ( 1 , 1 ) .

Network Contribution Games (c.c. upper bound) Notation their tribe. Say player i is witness to the edge e they invest in. Witnesses 16 u i ( s ) is the utility player i gets in s ; u τ i is the same for w e ( s ) is the utility edge e pays to its endpoints. Lemma (AH ’12): There exists an optimum s ∗ where every player’s strategy is tight: the whole budget goes into one edge. of e : W s ∗ ( e ) ⊆ P .

Lose at most 2 u i s u j s , when i j and their partners in s Gain at least w e s u i s u i s i s j s u i s 2 u i s u j s w e s Network Contribution Games (c.c. upper bound) i j i j . , when are all the same tribe. tribe would stand to lose from both endpoints deviating to s . at least one endpoint’s Stable against bilateral deviations 17 Pick a Nash equilibrium s , and consider each edge e = { i , j } in turn. If | W s ∗ ( e ) | = 2: .

Lose at most 2 u i s u j s , when i j and their partners in s Gain at least w e s u i s 2 u i s u j s w e s Network Contribution Games (c.c. upper bound) j . 17 i , when are all the same tribe. Pick a Nash equilibrium s , and consider each edge e = { i , j } in turn. If | W s ∗ ( e ) | = 2: Stable against bilateral deviations ⇒ at least one endpoint’s tribe would stand to lose from both endpoints deviating to s ∗ . u τ u τ i ( s ∗ i ; s ∗ i ( s ) ≥ j ; s − i , j ) .