Abstract
In the early 20th century, Pigou observed that imposing a marginal cost tax on the usage of a public good induces a socially efficient level of use as an equilibrium. Unfortunately, such a “Pigouvian” tax may also induce other, socially inefficient, equilibria. We observe that this social inefficiency may be unbounded, and study whether alternative tax structures may lead to milder losses in the worst case, i.e. to a lower price of anarchy. We show that no tax structure leads to bounded losses in the worst case. However, we do find a tax scheme that has a lower price of anarchy than the Pigouvian tax, obtaining tight lower and upper bounds in terms of a crucial parameter that we identify. We generalize our results to various scenarios that each offers an alternative to the use of a public road by private cars, such as ride sharing, or using a bus or a train.
Keywords
- Price of anarchy
- Pigouvian tax
- Public good
- Ride sharing
Supported by the European Research Council (ERC) under the European Union’s Horizon 2020 research and innovation programme (grant agreement No 740282).
Supported by the Israeli Smart Transportation Research Center (ISTRC).
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Notes
- 1.
The individual cost function is also the inverse function of the demand function. For every \(q\in [0,1]\), q fraction of the population have disutility at least \(\alpha (q)\) for not using the public good (and using the alternative instead).
- 2.
We also study a variant where only those that use the public good incur the cost of l(q), see Sect. 5.
- 3.
Under appropriate continuity assumptions.
- 4.
If no equilibrium exists then we define the price of anarchy to be 1. An equilibrium always exists if t is continuous. None of our results rely on non-existence of equilibrium.
- 5.
It is not difficult to see that the problem becomes trivial if the demand is fixed and the tax function may depend on it.
- 6.
If the optimal social cost is zero then the price of anarchy is defined to be infinity, unless every equilibrium has zero social cost, in that case the PoA is defined to be 1.
- 7.
The assumption that \(\kappa <1\) corresponds to the externality of a passenger in a private car being larger than that of a passenger riding a shared car.
- 8.
This formulation is general enough to capture ride sharing vehicles that impose larger load on the road than private cars, as long as the per-passenger load is at most 1: for example, a minibus with 20 passengers that have total load of 2 (like two private cars) has a per-passenger load of 1/10.
- 9.
The assumption that \(\tau <1\) corresponds to the natural assumption that the toll charged to a passenger of a private car is larger than the toll imposed on a passenger in a shared car.
- 10.
Note that the total toll paid by all carpool passengers might not necessarily be equal to the toll a private car is charged.
- 11.
The assumption that no congestion is created by an infinite-size bus is ideal, and clearly not very realistic. A more nuanced model has finite-capacity ride sharing vehicles that do increase road congestion.
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Babaioff, M., Mundel, R., Nisan, N. (2022). Beyond Pigouvian Taxes: A Worst Case Analysis. In: Feldman, M., Fu, H., Talgam-Cohen, I. (eds) Web and Internet Economics. WINE 2021. Lecture Notes in Computer Science(), vol 13112. Springer, Cham. https://doi.org/10.1007/978-3-030-94676-0_13
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