Why You Should Charge Your Friends for Borrowing Your Stuff

Why You Should Charge Your Friends for Borrowing Your Stuff

Kijung Shin,  Euiwoong Lee,  Dhivya Eswaran,  and Ariel D. Procaccia
Carnegie Mellon University, Pittsburgh, PA, USA
{kijungs, euiwoonl, deswaran, arielpro}@cs.cmu.edu
Abstract

We consider goods that can be shared with -hop neighbors (i.e., the set of nodes within hops from an owner) on a social network. We examine incentives to buy such a good by devising game-theoretic models where each node decides whether to buy the good or free ride. First, we find that social inefficiency, specifically excessive purchase of the good, occurs in Nash equilibria. Second, the social inefficiency decreases as increases and thus a good can be shared with more nodes. Third, and most importantly, the social inefficiency can also be significantly reduced by charging free riders an access cost and paying it to owners, leading to the conclusion that organizations and system designers should impose such a cost. These findings are supported by our theoretical analysis in terms of the price of anarchy and the price of stability; and by simulations based on synthetic and real social networks.

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Why You Should Charge Your Friends for Borrowing Your Stuff


Kijung Shin,  Euiwoong Lee,  Dhivya Eswaran,  and Ariel D. Procaccia Carnegie Mellon University, Pittsburgh, PA, USA {kijungs, euiwoonl, deswaran, arielpro}@cs.cmu.edu

1 Introduction

Social networks are known to play an important role in the everyday choices people make. In particular, a significant body of work studies the network effect, in which there are payoffs from aligning one’s decision with those of others [????]. For example, direct payoffs arise when friends or collaborators use compatible technologies instead of incompatible ones, that is, the game rewards coordination.

Our work considers the purchase of shareable goods, which, in a sense, gives rise to a certain type of anti-coordination game. Indeed, buying such a good yields a benefit only when no friend buys the good, since otherwise free riding is possible. An example that would be familiar to most parents is seldom-used baby gear, such as portable cribs, which are frequently borrowed by friends or friends of friends; similar examples include ski gear and hiking equipment. Expensive lab equipment provides a more pertinent example: Confocal laser scanning microscopes, or polymerase chain reaction (PCR) machines, are typically bought by one investigator and used by collaborators. In the realm of AI, one can imagine a multi-agent system populated by heterogeneous software agents that interact and share special computational resources, e.g., a high-end graphics processing unit (GPU) for particularly demanding image processing tasks.

To examine incentives to buy shareable goods, we devise game-theoretic models where each node on a network decides whether to buy a good that is shareable with -hop neighbors (i.e., nodes within hops from an owner), or free ride. Specifically, the good in question is non-excludable and non-rivalrous in that no -hop neighbor can be excluded from use, and use by a neighbor does not reduce availability to others. Note that the goods in the examples given above are (essentially) non-rivalrous, as any single person (or agent) requires the good only from time to time.

We find that social inefficiency, specifically excessive purchase of the good, occurs in Nash equilibria. Moreover, the social inefficiency decreases as increases and thus a good is shared with more people. Finally, charging free riders an access cost and paying it to owners also significantly reduces the social inefficiency. We support these findings both theoretically and experimentally.

Compared with previous work on shareable goods on a network, discussed in Section 5, our contributions are as follows:

  • Efficiency Analysis of Equilibria : We provide worst-case analysis of the efficiency of equilibria, in terms of the price of anarchy and the price of stability.

  • Simulation on Real-world Networks: The simplicity of our model allows us to measure social inefficiency on real-world social networks, with thousands of nodes, through simulations of best-response dynamics.

  • Mechanism Design: We analyze the effects of access costs on social inefficiency and suggest an appropriate cost for minimizing social inefficiency.

The rest of the paper is organized as follows. In Section 2, we define shareable goods games on a network. In Section 3, we give a theoretical analysis of the efficiency of equilibria. In Section 4, we present simulation results. After discussing related work in Section 5, we draw conclusions in Section 6.

2 Our Models

In this section, we formally define two game-theoretic models of the purchase of shareable goods on a network.

2.1 Shareable Goods Game (SGG)

Consider an undirected network where The players of the game are nodes in . Each node decides whether to buy a good or not. The strategy of node is denoted by , where denotes the strategy set of node . If node buys the good then , and otherwise (only pure strategies are considered). Given any strategy profile , we use to denote the strategies taken by all nodes but . Then, is also denoted by . The price of a good is (), which is identical for all nodes. A node gets benefit () by having access to a good and otherwise. Each node can access a good if it buys the good itself or has at least one node who buys the good within hops. We assume that having access to multiple goods does not increase the benefit of a node and that being accessed by multiple nodes does not decrease the benefit derived from a good (non-rivalry).

state conditions utility (i.e., )
buy ()
free ride ()
no access
Table 1: Utility in an SGG.
state conditions utility (i.e., )
buy
rent ,
no access ,
Table 2: Utility in an SGG-AC.

In this setting, the utility of node under strategy profile depends on the strategies of its -hop neighbors (i.e., the set of nodes within -hops from including itself), as given in Table 1. Note that each node gets the highest utility when it free rides and the second highest utility when it buys the good. Each node gets the lowest utility when neither the node nor its -hop neighbors buy the good.

SGG extends the best-shot game [?], which is equivalent to SGG if , by considering not only direct but -hop neighbors. SGG-AC, discussed in the following subsection, further extends SGG by considering access costs.

2.2 Shareable Goods Game with Access Costs (SGG-AC)

In this subsection, we extend the game defined in the previous section to a game we call the shareable goods game with access costs (SGG-AC), where each free rider has to pay an access cost. We focus on the differences from an SGG.

The strategy set of each node is , its -hop neighbors including itself. If node buys a good then , and if node does not buy a good but wants to access a good bought by node then . If for , and node actually buys a good (i.e., ) then node derives benefit from the good at the expense of paying an access cost of () to node . The followers of node , the set of nodes who want to access the good bought by node , are denoted by . Then, the utility of node under strategy profile in an SGG-AC is given in Table 2. Define the follower threshold ; for ease of exposition we assume that is not an integer. If node has at least followers (i.e., ), it has the highest utility when it buys a good (i.e., ). Otherwise, renting a good is preferred. Each node has the lowest utility when it is not accessing any good.

3 Analysis of Equilibria

5

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(a) Not NE

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6

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(b) NE ()

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(c) NE ()
Figure 1: Example strategy profiles in an SGG when . (a) is not an NE since each of node 3 and node 6 would be better off not buying. Between the NEs, (c) leads to a lower social cost than (b).

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(a) Not NE

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(b) NE ()

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(c) NE ()
Figure 2: Example strategy profiles in an SGG-AC when and . Arrows indicate who accesses whose products. (a) is not an NE since node 6 is better off not buying. Between the NEs, (c) leads to a lower social cost than (b).

In this section, we define equilibria in the games described in Section 2. Then, we analyze the efficiency of the equilibria in terms of price of anarchy (PoA) and price of stability (PoS).

3.1 Definition and Existence of Equilibria

We use the ubiquitous concept of Nash equilibrium (NE) as our solution concept. We first formally define it.

Definition 1 (Nash Equilibrium).

A strategy profile is a Nash equilibrium (NE) if no node can increase its utility by changing its strategy given the strategies of the other nodes, i.e.,

Figures 1 and 2 give examples of NEs in an SGG and an SGG-AC, respectively, with explanations. Note that a strategy profile is an NE in an SGG if and only if the set of owners is a -independent dominating set [?], i.e., any two owners have distance at least and every node has distance at most to some owner. Theorem 1 states that an NE always exists in both games.

Theorem 1 (Existence of Nash Equilibria).

An NE exists in any SGG and SGG-AC.

Proof. Given , and , the following procedure gives an NE for any SGG and for any SGG-AC. Choose an arbitrary node in the graph, let buy a good (, ), and for each node within hops from , let follow (, . Delete and all nodes within hops from , and repeat until there is no node left in . At the end, every node either buys a good or accesses its -hop neighbor’s.

Each node that accesses its -hop neighbor’s good cannot increase its utility by buying a good since the utility of accessing its -hop neighbor’s is greater than that of buying (and no one follows). Each node that buys a good also cannot increase its utility by following another node because the procedure ensures that there is no node that buys a good and is within distance from . Therefore, and are NEs for the given SGG and SGG-AC, respectively. ∎

3.2 Social Inefficiency in Equilibria

We now turn to the analysis of NEs in our games. It is important to note that a node that does not access any good can increase its utility by buying a good, without decreasing the utilities of the others in both of our games (see Tables 1 and 2). Thus, if we let be the set of strategy profiles where every node accesses a good and thus gets benefit 111In SGG, .
In SGG-AC, , ( or ( and )).
, then all NEs belong to . Due to the same reason, every socially optimal strategy profile (i.e., strategy profile maximizing social welfare ) belongs to . Therefore, to define PoA and PoS, we only need to consider the strategy profiles in . Since all strategy profiles in have the same sum of benefits, we can compare them simply by their social cost, which is proportional to the number of nodes buying a good (see Definition 2). Importantly, access costs in SGG-AC cancel out (they are paid by some players to others) and do not affect the social cost.

Definition 2 (Social Cost).

Given a graph , the social cost of a strategy profile is the sum of prices paid by the nodes, i.e.,

in SGG in SGG-AC
( if )
( if )
Table 3: Summary of our analysis of efficiency of equilibria.

The price of anarchy (PoA) is defined as the social cost of the worst NE divided by minimum social cost (see Definition 3) and the price of stability (PoS) is defined as the social cost of the best NE divided by minimum social cost (see Definition 4). Large PoA and PoS indicate that NEs are socially inefficient.

Definition 3 (Price of Anarchy).

Given a graph with nodes, the price of anarchy (PoA) is defined as

Definition 4 (Price of Stability).

Given a graph with nodes, the price of stability (PoS) is defined as

.

.

.

.

.

.

-

-

(a) Best NE in SGG

.

.

.

.

.

.

-

-

(b) Best NE in SGG-AC
Figure 3: An example of social inefficiency in an SGG. Assume . Arrows indicate who accesses whose products. In this example graph, the best NE in an SGG is (a), whose social cost is . In an SGG-AC (with ), however, the best NE is (b), whose social cost is , equal to the minimum social cost.

In Table 3, we summarize the results of our worst-case efficiency analysis. That is, we analyze the two measures in the worst case over all graphs. As stated in Theorem 2, both the PoA and PoS in SGGs are in the worst case. That is, not only worst NEs but also best NEs can be severely inefficient, with social cost as high as times the optimum. Figure 3(a) shows an example of such inefficiency for , where even the best NEs in the SGG have social cost , while the minimum social cost, in Figure 3(b), is .

Theorem 2 (PoA and PoS in SGG).

and in SGG are both in the worst case.

Proof. For the upper bound on , fix an arbitrary NE . Let , and for each node that buys a good, consider , the set of nodes within distance from (called a ball around ). Since each pair of nodes that buy a good are at distance at least from each other, these balls are pairwise disjoint.

Call a ball big if it has at least nodes, and small otherwise. If a ball is small, is in a connected component with less than nodes, and there is no other node that buys a good in that component. Let be the number of connected components in ; there are at most big balls and small balls. The number of nodes that buy a good is equal to the number of balls, which is at most .

Since the optimal social cost is at least , the ratio between the social cost of and the optimum is at most .

For the lower bound on , given integers , consider a tree where there are two center nodes and , and simple paths with nodes. For of them (called left arms), one of two endpoints is connected to . For the other of them (called right arms), one of two endpoints is connected to . Finally, and are connected. Figure 3 shows such a graph with and . It is easy to see that the optimal social cost is at most , since if and buy a good, all nodes can access at least one good.

We claim that any NE has social cost at least . Fix an NE . In , either or does not buy a good, since they can access each other’s goods. Without loss of generality, suppose that does not buy a good. Consider a left arm, specifically the endpoint of the arm not connected to . The only nodes within distance from this endpoint are and the nodes in the same arm. Since does not buy a good, there must be a node in the same arm who buys a good. This argument holds for each left arm, so at least nodes buy a good, establishing the claim.

Since the optimal social cost is at most and any NE has social cost at least , . Since , the theorem holds. ∎

Intuitively speaking, the main reason for the inefficiency of NEs in SGGs is that high-degree nodes (i.e., nodes with many -hop neighbors) are less likely to buy goods even when many neighbors can benefit from goods bought by high-degree nodes. Indeed, high-degree nodes are more likely to have a neighbor buying a good, and thus choose to free ride.

To incentivize high-degree nodes to buy goods, we can force their neighbors who access the good to pay an access fee to the node — as we do in SGG-ACs. In Figure 3(b), for example, the two high-degree nodes in the center still buy goods, even when they can free ride, since they receive access fees from or more followers, minimizing social cost.

This improvement through access costs is formalized and generalized in Theorem 3, where we show that the worst-case PoS in SGG-ACs is , which is significantly smaller than in SGGs. In particular, if , then the PoS in SGG-ACs is , i.e., the social cost in the best equilibria is always optimal even in the worst case. Among values satisfying the condition, the largest one (i.e., ) is preferred to minimize the PoA, which is inversely proportional to , as shown by Theorem 4.

Theorem 3 (PoS in SGG-AC).

in SGG-ACs is in the worst case. In particular, it is (i.e., there are guaranteed to be socially optimal equilibria) if ).

Proof. For the upper bound, given , , and , let be a smallest set of the nodes such that for each node , there is a node such that and are within distance from each other (i.e., is the optimal social cost). Consider a strategy profile where each node in buys a good, and all other nodes access a good of a node in such that for each , the set of nodes that access ’s good induces a connected subgraph. Call a node an owner if it buys a good. An owner is called a rich owner if , and a poor owner otherwise. That is, an owner is a rich owner if and only if is at least the utility of renting a good. Note that is not an NE if and only if there is a poor owner who can access a good of another owner.

From , we show how to construct an NE whose social cost is at most . If there is a poor owner who can follow another owner , let follow . For each node who previously followed , if it can follow another owner, let it follow that owner. Call a node underprivileged if it previously followed but cannot follow any other owner. Scan the list of underprivileged nodes sequentially. When is considered, let be an owner (call it a new owner) and be followed by all still underprivileged nodes who can follow (and remove them from the underprivileged nodes list). At the end of this loop no node is left underprivileged. Repeat until no poor owner can follow another owner .

One of the invariants of this procedure is that any new owner can never access a good of any other owner — became a new owner since it could not access a good of any other owner, and, subsequently, any node cannot become a new owner, as it can follow , i.e., . Therefore, this procedure always terminates, after having at most owners follow other owners. The final strategy after termination is an NE since there is no underprivileged node and no poor owner who can follow another owner.

To bound the number of new owners, note that when an owner deviated to follow another owner , among ’s previous followers including (call them ), at most new owners can be created. This is because (since was a poor owner) and if we let , and consider the ball around each new owner , and all balls are pairwise disjoint (all new owners are at distance at least from each other). The deviation of creates at most new owners. Therefore, the number of owners in the final NE is at most . If , so the resulting NE is a social optimum. The same conclusion holds when since one deviation creates at most one new owner.

For the lower bound of , for any integers such that is an integer, build the same tree as in the proof of Theorem 2. As before, the optimal social cost is , and the social cost at NEs is at least . To see why the latter claim holds, note that (as before) and cannot simultaneously be owners because the total number of nodes other than and is , so that at least one of and must be a poor owner. Using the same argument as before, the number of owners is at least , and . ∎

Theorem 4 (PoA in SGG-AC).

in SGG-ACs is in the worst case.

Proof. See Appendix A. ∎

Note that PoA in SGG-AC has the same order as PoA in SGG if .

4 Experiments

5 Related Work

6 Final Words

Acknowledgements

We thank Prof. Christos Faloutsos for fruitful discussions.

References

Appendix A Proof of Theorem 4

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