Paradoxical examples of social networks games with product choice
aa r X i v : . [ c s . G T ] A ug Paradoxical examples of social networks games with product choice
Raskin M.A. ∗ Nikitenkov N.S. † September 22, 2018
Abstract
Paradox of choice occurs when permitting new strategies to some players yields lower payoffs for all playersin the new equilibrium via a sequence of individually rational actions.We consider social network games. In these games the payoff of each player increases when other playerschoose the same strategy.The definition of games on social networks was introduced by K. Apt and S. Simon. In an article writtenjointly with E. Markakis, they considered four types of paradox of choice in such games and gave examples ofthree of them. The existence of paradoxical networks of the fourth type was proven only in a weakened form.The existence of so-called vulnerable networks in the strong sense remained an open question.In the present paper we solve this open question by introducing a construction, called a cascade, and use itto provide uniform examples for all four definitions of paradoxical networks.
One of the topics of interest in game theory is the paradox of choice. In formal game theory this name appliesto a situation where permitting new strategies to one or more players leads to some rational selfish decisions andeventually worsens situation (yields lower payouts) for all players in the new Nash equilibrium.Perhaps the most famous paradox of that type is Braess’s paradox [1]. In this game many players try to getfrom one node of a graph to another in a graph with edges having different bandwidths. It turns out that addingadditional edges can lead to increased transit time for all players.In the present paper we consider another subclass of games on graphs, the so-called social network games. Inthese games the payoff of each player increases when other players choose the same strategy. We will define thegame rigorously later.The definition of games on social networks was introduced in the article [2] by K. Apt and S. Simon. Theyhave obtained a number of theoretical results (in particular, it was shown that the problem of finding a Nashequilibrium in a social network game in the general case is NP-complete). In a later article [3], written jointly with ∗ Aarhus University, CS Dept. email: [email protected] , [email protected] . The work was partially supported by RFBR grant16-01-00362. The author acknowledges support from the Danish National Research Foundation and The National Science Foundationof China (under the grant 61361136003) for the Sino-Danish Center for the Theory of Interactive Computation and from the Centerfor Research in Foundations of Electronic Markets (CFEM), supported by the Danish Strategic Research Council. † Moscow State University, Mech. and Math. Dept.
1. Markakis, the authors considered four hypothetical types of paradox of choice in games on social networks andgave examples of three of them. The existence of paradoxical networks of the fourth type, the so-called vulnerablenetworks, was proven only in a weakened form. The question of the existence of vulnerable networks in the strongsense was left open. A vulnerable network always provides an example of paradox of choice as defined above.In the present article we solve the open problem of Apt, Markakis and Simon. We introduce a construction,called a cascade, and use it to provide uniform examples for all four definitions of paradoxical networks. Inaddition, we use cascades to construct a number of other examples, which also can be considered paradoxical,though in a slightly different sense.
Let V = { , . . . , N } be a finite set of nodes. We consider a directed graph with weighted edges G = ( V, E, ω )with non-negative edge weights: ω ij >
0. We assume that the weight of a missing edge is zero. Let Π be a finiteset of possible individual strategies or products . Let P : V → Π be a function describing available products(i.e. allowed strategies) for every player. Let θ be a threshold function . For each player i ∈ V and each product t ∈ P ( i ) the value θ ( i, t ) > use price . Definition 1. A social network (or simply a network) is a set S = ( G, Π , P, θ ) . Definition 2.
A network S ′ is called an expansion of the network S if it is obtained by allowing a singleadditional product for a single node. The network S is called a contraction of the network S ′ . Now we consider a strategic game with each node of the social network S being an independent player. Allplayers simultaneously make a choice between use of one of the available products t i ∈ P ( i ) or refusal to use anyproduct (we will call this choice t ). Thus, for each player i ∈ V we have defined a set of strategies S i = P ( i ) ∪{ t } .We will use the notation G ( S ) to denote the strategic game based on S . We will call a strategy profile of allplayers s ∈ Q i ∈ V S i a position of the game G ( S ). In a position s , let s i denote the individual strategy for player i in s , and let N ( t ), where t ∈ Π, denote the set of all players who have selected the product t .Each player will pay the price of the selected product and will receive utility proportional to the incoming linkweight whenever an acquaintance has chosen the same product. That is, we define the payoff function p i in thefollowing way: p i ( s ) = , if s i = t ( P j ∈ N ( t ) ω ij ) − θ ( i, t ) , if s i = t ∈ P ( i ) Definition 3.
Let two positions, s and s ′ differ only by the individual strategy for player i : ∀ j = i : s j = s j . Let p i ( s ) < p i ( s ′ ) . Then we say that s ′ is obtained by an individual improvement of s .If a position s allows no individual improvements, the position s is a Nash equilibrium for the game G ( S ) . n individual improvement path is a sequence of positions, in which each next position is an individualimprovement on the previous one. Given two states s and s ′ in a game G ( S ) we write that s > s ′ if ∀ i ∈ V : p i ( s ) > p i ( s ′ ). Definition 4.
A network S is called vulnerable , if for some Nash equilibrium state s in G ( S ) there exists anexpansion network S ′ such that each individual improvement path in G ( S ′ ) starting at the state s leads to a state s ′ < s , which is a Nash equilibrium in both G ( S ) and G ( S ′ ) . Definition 5.
A network S is called fragile , if for some Nash equilibrium state s in G ( S ) there exists anexpansion network S ′ such that each individual improvement path in G ( S ′ ) starting at the state s is infinite. Definition 6.
A network S is called ineffective , if for some Nash equilibrium state s in G ( S ) there exists acontraction S ′ such that each individual improvement path in G ( S ′ ) starting at the state s leads to a state s ′ > s which is a Nash equilibrium in both G ( S ) and G ( S ′ ) . Definition 7.
A network S is called unsafe , if for some Nash equilibrium state s in G ( S ) there exists acontraction S ′ such that any individual improvement path in G ( S ′ ) starting at the state s is infinite. Theorem.
There are networks of all four types described.
This theorem has been partially proved in [3]. The authors give isolated examples of fragile, inefficient andunsafe networks, and prove the existence of vulnerable networks in a weaker sense, namely for social games inwhich players are not allowed to refuse to use any product.Before giving the proof of the theorem, we will consider a network design that we call a cascade . We will useit to construct an example of a vulnerable network in the original sense and also a uniform series of examples forthree other paradox types.
A cascade is a social network with three products. It can include an arbitrarily large number of playersand always has two non-trivial equilibriums states (the choice can be affected by incoming edges). The transitionfrom the first equilibrium to the second one leads to a severe reduction in the individual payoff of each player.Moreover, this payoff difference grows proportionally to the size of the cascade and can reach arbitrarily largevalues.In the situation when a cascade is built into a larger social network in the form of a subgraph, changes in thepayoff can also apply to some of the vertices outside the cascade. In a way, the cascade is a “happiness machine”that can be “turned on”, making all the players in the system arbitrarily happy, and “switched off”, deprivingthem of the additional utility.We will now describe the construction of the cascade.In a cascade there are three products denoted by A, B and C. In the diagrams below, we enumerate availableproducts for each player directly inside the corresponding node of the graph.3 .1 Types of players
The cascade includes two types of players called “humans” and “spirits”.Each “human” player has a choice of two different products. These products correspond to the two differentnon-trivial equilibriums. Of course, there are six possible subtypes of human players: AB, AC, BA, BC, CA, CB.Every cascade includes an equal number of players of all six subtypes.The number of players of one subtype will be denoted by n . The total number of human players is 6 n .A spirit player has only one product available, which he will use in both equilibriums of the cascade. Thecascade uses six spirits: two for each of three products.The thresholds for all players are equal and close to zero. We’ll use θ to denote them. In every cascade, there are edges of three different types.The control edges have the largest weight. In a cascade, each player has at most one outgoing and oneincoming control edge. The design of a cascade is such that each player always follows the incentives set by theincoming control edge whenever possible (i.e. when the corresponding product is available). In the diagrams thecontrol edges will be shown by solid black arrows. Each control edge has the same weight c .The inclination edges determine the behavior of a player when the corresponding control edge incentivisesthe use of an unavailable product. Each human player has exactly one incoming inclination edge from one of sixspirits. The spirits themselves do not have incoming inclination edges.Since the spirits cannot change their chosen product, each inclination edge gives its target player a constantpreference for using some product. In the diagrams we will emphasise the corresponding product if it is available.All inclination edges have the same weight i < c .The remaining edges inside a cascade are called emotional edges. Their individual weight is comparativelysmall, however, every player in the cascade has a lot of incoming emotional edges, so in total they have a significanteffect. However, incoming emotional edges never influence a player’s strategic choices. More precisely, for eachplayer in the cascade the following property always holds. Definition 8. (emotional invariant)
Denote by E ( v, α ) the number of incoming emotional edges for the player v rewarding the choice of the product α . We will say that the emotional invariant holds for the player v , if foreach pair of products ( α, β ) available to the player v , | E ( v, α ) − E ( v, β ) | . Note that there are no restrictionsfor the value E ( v, γ ) when the product γ is not available to the player v . Thus, all incoming emotional edges for each player are always balanced between all the available products (thedifference is no more than a single edge). Switching the emotional edges to reward the choice of the unavailableproducts allows to reduce the eventual payoffs.Emotional edges will not be indicated on the diagrams. We will only describe their layout in the text. All theemotional edges have the same weight, we’ll call it e . The following conditions will hold: e > , e < i, e < c, e < c − i ,but en > c + i . 4 .3 Components of the cascade Each stimulus consists of isolated an pair of identical spirits, with control edges in both directions. Thiscoupling ensures that the spirits will never refuse to use the product available to them.
A-stimulus B-stimulus C-stimulus
A A B B C CThe only purpose of the spirits in the stimuli is to serve as the sources of inclination edges to human players.Recall that each human player has exactly one incloming inclination link, which determines its behavior in thecase when following the control link is impossible.
Each rank is a chain of human players. All rank members share one available product; each of them hasanother available product that alternates between the remaining two options. Rank members are connected bycontrol edges along the rank.All players in the line, except the first, have the inclination towards the second (alternating) available product,and the first human has an inclination towards the first (common) available product. Common product for aline of players will be called the main product for this line, and the two alternating products will be called secondary . We will refer to the ranks by mentioning their main products.A cascade uses a single rank of each type with 2 n players in each. The figure below shows an example of anA-rank: A-rankA
B A C A B A C ... A B A C Definition 9.
We say that rank has emotional influence on player v , if each player in the rank is connectedto v with one outgoing emotional edge. The rank can have an emotional influence on several players, both insideand outside of the cascade. Each player in the rank, in turn, is under external emotional influence (subject to emotional invariant). Inaddition, the first player in the rank has an incoming control edge from some external player and the last playerhas an outgoing control edge to some external player (or players). The players inside a rank have no other externalconnections.Consider the possible states of a rank in the assumption that the emotional invariant set forth above is notviolated (in other words, incoming emotional edges will not affect the strategic choices of players).
Lemma 1. (on rank states)
Assume that the emotional invariant holds for every player in a rank.Also assume that initially either everyone uses the main product or everyone uses the secondary product.Then the choice of the first player in the rank between the main and the secondary product becomes the choiceof the entire row as a result of the only possible chain of individual improvements.
Proof.
First, note that if the first player in the line uses the main product (at least because of its inclinationedge), under the influence of the control edges all the rank will also be forced to use the main product (despitetheir inclinations).Now consider a chain of improvements if the first player will change its strategy in favor of a secondary product.Then the next player will not be able to submit to the control edge and so will also change their choice in favor ofthe secondary product under the effect of incoming inclination link. This in turn will make it impossible for thethird player to follow the incoming control edge and so on. As a result all the rank will consistently change theirchoice in favor of secondary products. Obviously, this is the only possible chain of improvements in the describedcase.The converse is also true: if you remove the external influence on the first player, then the only possible resultof a chain of improvements is for all the rank to return to the main product. (cid:4)
We will call the first state of a rank the of the main product by all the players, and the second state ofa rank , respectively, the use of secondary products.Now we prove the main property of a rank:
Lemma 2. (on maintaining emotional invariant)
Let a rank with the main product α and the secondary products β and γ have an emotional influence on an external player v . Then in both equilibrium states of the rank and atany step during the chain of improvements leading from one to another, | E ( v, β ) − E ( v, γ ) | . Proof.
In the first state of the rank none of its constituent players uses the products β and γ and the specifiedproperty is trivially true. In the second state of the rank exactly half of the players use the product β , and theremaining half use the product γ , so E ( v, β ) = E ( v, γ ). During the transition from the first state to the secondone, players in line sequentially switch from α to β , then from α to γ , and hence the magnitude of E ( v, β ) and E ( v, γ ) in the queue increases by one. Similarly, during the transition from the second state of the rank to the6rst one, the players sequentially switch from β and γ to α , hence the value of E ( v, β ) and E ( v, γ ) in the queuedecreases by one. In both cases | E ( v, β ) − E ( v, γ ) | stays less than or equal to 1 all the time. (cid:4) Note that if the player v can use only the products β and γ or only one of them (and no other productsare available), the emotional invariant property means that the influence of the emotional edges from the rankis always balanced between the available products. In such a case in the second state the rank gives additionalutility ne to the player v compared to the first state of the rank. A-stimulus B-stimulus C-stimulus
A A B B C C
C-rank A-rank B-rankC AC B C A C B ... C A C B A BA C A B A C ... A B A C B CB A B C B A ... B C B A The cascade consists of three different stimuli and three different ranks connected by control edges as shownin the figure.Each rank has emotional influence on all the players who don’t have access to its main product. So, the A-rankhas an emotional influence on all human players types BC and CB, and the spirits of types B and C.
Lemma 3.
In the above construction of the cascade the state of the first rank is translated to the second and thirdranks in the result of the only possible chain of individual improvements. The individual payoffs of each player inthe second state are higher than the corresponding payoffs in the first state by at least ( ne − c − i ) . Proof.
First note that for each player in the cascade (according to
Lemma 2 ) the emotional invariant holds.Therefore, if the first rank is in the first state, the second rank will not be able to follow incoming control edge,and (
Lemma 1 ) will also switch to the first state. Similarly, it will be impossible for the third row to follow theincoming control edge, so it will also adopt the first state.In this case, all emotional edges will push players to the selection of products available to them.7f the first rank moves into its second state, the second rank will be able to follow incoming control edge, and(
Lemma 1 ) also so it will move into its second state, making the third rank also switch to the second rank.In this case, due to the emotional edges all the human players receive extra payoffs of size ne , and all the spiritplayers get an additional gain of 3 ne that obviously outweighs the gains from control and inclination edges.Thus, individual payoffs of all the players in the second state of the cascade are greater than in the first stateof the cascade by at least ( ne − c − i ). (cid:4) Note that since n can be chosen arbitrarily high, the potential difference of the payoffs is unlimited. We now formulate the properties in cascade the form of a theorem:
Theorem. (on the properties of the cascade)
The design of the cascade has the following properties:1. The cascade has one incoming control edge and one outgoing control edge.2. If a cascade is used as a source of inclination edges to external players, these edges incentivise selection ofthe same products, regardless of the state of the cascade.3. There is an equilibrium state of the cascade when the incoming edge incentivises the use of the product Aand the cascade is in its second state. The outgoing edge in this situation also incentivises the use of theproduct A.4. There is also an equilibrium state of the cascade when the incoming edge incentivises the use of any otherproduct and the cascade is in the first state. The outgoing edge in this situation incentivises the use of theproduct B.5. If the incoming edge switches when the cascade is in one of the two aforementioned equilibrium states, thecascade switches to the other state following the only possible individual improvements sequence.6. In the second state the cascade gives additional emotional payoff to all its players.7. The individual payoffs of all the players in the second state of the cascade are higher than in the first stateof the cascade at least by ( ne − c − i ) .8. The emotional influence of the cascade can be extended to external players having access to one or twoproducts from the set { A,B,C } and no other products. In respect of these players the emotional invariantwill hold and they will receive additional emotional payoff in the amount of at least ne when the cascade isin the second state, and no emotional payoff, when the cascade is in the first state. Proof.
The cascade device and
Lemmas 1, 2 and 3 . (cid:4) These properties will be used when using the cascade as a subgraph in a larger network.8 .6 Choosing parameters
Above we have formulated conditions on the numerical parameters of the cascade: θ > e > i > e ; i > θ ; c > i + e ; n > c + ie ;These conditions can be met by selecting each parameter large enough. In order to demonstrate the existenceof the cascade in principle, it is enough to consider θ = 1 , e = 1 , i = 3 , c = 5 , n = 10.We can also determine the minimum size of the cascade that meet the specified conditions. Lemma 4.
The minimum number of players in the cascade is 30.
Proof.
Combining inequality c − i > e and 2 i > e we get c + i > e , and hence the compliance is impossiblewhen n < n = 4 a cascade exists, for example, with these values: θ = 0 . , i = 0 . , c = 0 . , e = 0 . · (cid:4) Now we return to the proof of the main theorem about the paradoxical networks. A1 Cascade B C Proof.
We add two players to a cascade as shown in the diagram (solid arrows denote control relations, dottedarrow pointing to the first player denotes adding an additional product). Note that the second external playerneeds the inclination edge from one of the C-spirits. In addition, both external player must be under emotionalinfluence from B- and A-ranks, respectively.Initial Nash equilibrium: the first player uses the product A, the cascade under the influence of the controledge is in the second state, the second player can’t follow the control edge, so according to his inclination he usesthe product C. Since the cascade is in the second state, all players (including the two additional players) get anadditional payoff due to the emotional edges. Because of this extra payoff the first player will not refuse to use theproduct A. He can’t obey incoming control edge, as the product C is not available. Thus, the network is indeedin equilibrium. 9ow allow the first player to use the product C. Emotional edges won’t affect the choice between A and C,so he will obey the incoming control edge and switch to the product C. This will lead to the transition of thecascade into the first state and the control edge will now force the second player to switch and use the product B.The control edge from the second player to the first becomes impossible to obey for the first player. Moreover, allthe incoming edges will reward choosing the unavailable product B. The first player will abandon the use of anyproduct to save on the threshold cost. This will not lead to any further changes, so the network will have reacheda new equilibrium with strictly lower payoffs for all players. Note that this position is also an equilibrium in theoriginal network.Thus, the constructed network is really vulnerable. (cid:4)
Note that in [2] it was proved that a vulnerable network can’t be built using less than three products. Thepresented example demonstrates that three products are sufficient. A Cascade B C Proof.
The only difference from the previous example is the inclination of the first player to use the product A.Due to that change, the first player will not refuse to use any of the products when the incoming control edge isimpossible to obey. He will return to the use of the product A instead. This would entail the return of the cascadein the second state, the transition of the second player back to product C and infinite repetition of the cycle.Thus, the constructed network is fragile. (cid:4)
Note that in the expanded network from the previous example, there exists a Nash equilibrium (total refusalof all players to choose any products), however, the only possible chain of individual improvements will not reachthat equilibrium.Also note that in this and the subsequent examples of endless cycles of individual improvements (without globalpayoff requirements), the main part of the cascade can be replaced by a single player of type A B . However, inthis case we would also need external sources of inclination edges. Cascade B A roof. In this example we need only one additional player, with inclination to A and under the emotionalinfluence of the C-ranks.Original equilibrium: the added player uses the product B. Therefore, the cascade is in the first state, all theemotional edges are “disabled” (i.e. they push players to an impossible choice), but the control edge holds thelast player in the state B. Thus, the network is indeed in an equilibrium.Now forbid the added player to use the product B. Under the influence of the inclination edge, he will beforced to use the product A. This switch will make the cascade transition into the second state. All participatingplayers will receive additional payoffs due to emotional edges.Note that the resulting position is an equilibrium, and would be an equilibrium in the original network. Indeed,after the cascade switches into the second state, the incoming control edge of the added player will push him tochoose A and not B. (cid:4) B C A Cascade B C Proof.
This example largely repeats the fragile network. An additional player of type BC prevents the infiniteloop. Obviously, forbidding this player to use the product C will destroy his effect on the system (because thenext player has no access to the product B) and start the infinite chain of individual improvements.It remains to verify that the initial network is in equilibrium. The first player has no incoming control edges,and therefore acts according to his inclination and selects the product C. The second player will be forced tochoose product C, causing a cascade to switch into the first state, and the third player under will choose theproduct B. From this position, no individual improvement is possible, and therefore it is indeed an equilibrium. (cid:4)
Now we can prove the following simple statement:
Theorem. (a very bad network)
There is a network that is vulnerable, fragile, inefficient and unsafe.
Proof.
It suffices to combine the examples for vulnerable, fragile, inefficient and insecure network into a singlegraph and distribute the emotional influence of the cascades from the first and the third examples to all externalplayers (including internal players of other cascades).Note that although in this case each player has incoming emotional edges from multiple cascades, emotionalinvariant still holds, since no two cascades change their state simultaneously. (cid:4) “Paradoxes” with the edge weight changes We now present some networks where analogous events (global unidirectional payoff change, infinite individualimprovement chain) happen when changing the edge weights.Such behaviour seems to be an interesting application of the cascade construction, regardless of whether it isconsidered paradoxical.
Cascade
Proof.
In this most simple example, it is enough to attach the outgoing control edge of the cascade as its ownincoming control edge. Note that in this case, the second state of the cascade is in equilibrium. If we reduce theweight of the loopback edge to zero, the cascade will return to the first state with the loss of emotional payoff forall the players. Note that the initial state of the cascade is an equilibrium for the original closed-loop network. (cid:4)
Note that the weight of the selected edges can be reduced to any value lower value i − e (in this case, theincoming edge will never affect the strategic choice of a player even if it accidentally acts in the same direction asemotional imbalance: the inclination will still be stronger). So the same example can be used for both removal ofan edge and for reduction of an edge’s weight to some positive value. The following examples will also use thisfact. Cascade B A Proof.
Here cascade is initially in the first state. The cascade keeps an additional player in the state B, which inturn allows the cascade to remain in the first state.Now if we set the weight of the cascade’s outgoing control edge to zero, the additional player will be leftwithout an incoming control edge and will go into the A state due to the inclination edge. The cascade will switchinto the second state, giving all participants an additional emotional payoff. Note that this configuration is anequilibrium for the original network. (cid:4) .3 Infinite loop when reducing edge weight C A Cascade B C Proof.
In this example we will use the same infinite loop design as for a fragile network. The equilibrium ismaintained by a dedicated control edge from the player using the product C. If this edge becomes zero-weight, aninfinite cycle of individual improvements will start. (cid:4)
CA1
Cascade B C Proof.
This example repeats the example of a vulnerable network, bu instead of allowing the product C to thefirst player, we add a control edge (marked on the diagram by a dotted line), making him switch to the productC, and remove the inclination for this product. (cid:4) A B Cascade
Proof.
In this example, the cascade is initially in its first state; however, increasing the weight of the dottedconnection from zero to c forces the second player to move to state A. The switch triggers the cascade transitioninto the second state. Note that the resulting configuration is an equilibrium even in the original network. (cid:4) .6 Infinite loop when increasing edge weight C A Cascade B C Proof.