Abstract
We use game theory to model a general participation game. The main problem we are interested in is how to achieve broad participation while aligning the incentives of all the participating agents. A consumer node is willing to invest some initial amount of money to get a set of networked nodes, alternatively agents, to participate in a desirable activity. The consumer, in this case a BGP speaker, desires to advertise itself; however, it may only communicate with its direct neighbors. Therefore, it must incentivize its neighbors to participate in further advertising its route, who then incentivize their neighbors to participate, and so on. We assume the commodity being traded to be the agent’s participation. In the resulting game, agents choose their offers strategically and they are rewarded by volume of sales. We prove the existence of equilibria for specific utility functions and simple network structures.
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Notes
- 1.
In this chapter, we use the term “control plan” to refer only to route prefix advertisements (not route updates) as we assume that the network structure is static.
- 2.
Metric-based policies could be modeled with HRP by fixing one of the players’ decisions. For example, fixing \(r_{ij}=r_{\mathrm{next}(R_{i})}-1\), ∀i,j results in hop count metric; or alternatively setting \(r_{ij}=r_{\mathrm{next}(R_{i})}-c_{i}\), where c i is some local cost to the node results in Least Cost Path (LCP) policy [59], etc.
- 3.
We abuse notation hereafter and we refer to the outcome with simply the strategy profile s where it should be clear from context that an outcome is defined by the tuple 〈s,r d 〉. Notice that a strategy profile may be associated with an outcome if we model r d as an action. We refrain from doing so to make it explicit that r d is not strategic.
- 4.
A preliminary estimate of this cost is shown by Herrin [83] to be $0.04 per route/router/year for a total cost of at least $6,200 per year for each advertised route assuming there are around 150,000 DFZ routers that need to be updated.
- 5.
This of course is an interesting question in its own right.
- 6.
This follows in the multi-stage game since a player at stage k will not offer rewards to its neighbors at stage l<k, i.e., rewards flow in one direction away from d. The outcome is necessarily a shortest-path tree since every player at stage k must choose its best route from the offers its received from neighbors at stage k−1.
- 7.
There is, however, a single mixed strategy equilibrium in which player 1 plays r 1=2 with probability \(\frac{2}{3}\) while player 2 plays r 2=1 with probability \(\frac{1}{2}\), yielding expected payoffs 6 and 5 for players 1 and 2, respectively.
- 8.
On the other hand, on complete d-ary trees, it may be shown that the function f(k)=Θ(k)=Θ(log d n) for d≥2 since the number of players, and hence δ i , grows exponentially with depth K. These growth results on the line graph and the tree seem parallel to the result of Kleinberg and Raghavan [97] (and the elaboration in [25]) which states that the reward required by the root player in order to find an answer to a query with constant probability grows exponentially with the depth of the tree when the branching factor of the tree is 1<b<2, i.e., when each player has an expected number of offsprings 1<b<2, while it grows logarithmically for b>2.
- 9.
Here player 1 has an advantage over player 2 and is threatening the latter to force a desirable outcome.
- 10.
When x<f(K−1), then g K (x)=f(K−1) by definition of f.
- 11.
Here g K (x) is the r 2 element in the solution.
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Appendix: Existence of g K (x)
Appendix: Existence of g K (x)
It is straightforward to show that g 3(x)=x+1 given the Bertrand competition of players 3 and 4 on the 3-stage ring. For K≥4 and for any r 1=x≥f(K−1),Footnote 10 \(\hat{\mathbf{s}}_{V_{\mathrm{even}}}\) is part of the solution to the following Integer Linear Program (ILP):Footnote 11
where β is a sufficiently large constant. The variables in the ILP above signify the actions of the players in the subgame G(h 2) while the constraints guarantee that all players compete while they have an incentive to do so knowing that each player may choose between competing or not. The constraints are constructed based on the definition of \(\hat{\mathbf{s}}_{V_{\mathrm{even}}}\) to make sure that players in V odd have no incentive to compete.
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Khoury, J.S., Abdallah, C.T. (2013). Participation Incentives in BGP. In: Internet Naming and Discovery. Signals and Communication Technology. Springer, London. https://doi.org/10.1007/978-1-4471-4552-3_8
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