Playing the Wrong Game: Smoothness Bounds for Congestion Games with Behavioral Biases

In many situations a player may act so as to maximize a perceived utility that is not exactly her utility function, but rather some other, biased, utility. Examples of such biased utility functions are common in behavioral economics, and include risk attitudes, altruism, present-bias and so on. When analyzing a game, one may ask how inefficiency, measured by the Price of Anarchy (PoA) is a?ected by the perceived utilities.
 The smoothness method [16, 15] naturally extends to games with such perceived utilities or costs, regardless of the game or the behavioral bias. We show that such biasedsmoothness is broadly applicable in the context of nonatomic congestion games. First, we show that on series-parallel networks we can use smoothness to yield PoA bounds even for diverse populations with di?erent biases. Second, we identify various classes of cost functions and biases that are smooth, thereby substantially improving some recent results from the literature.


INTRODUCTION
Game theory is founded on the assumption that players are rational decision makers, i.e. maximizing their utility, and that groups of agents reach an equilibrium outcome.However agents, either human or automated, often have bounded resources that prevent them from finding the optimal response in every situation.Further, human decision makers are prone to various cognitive and behavioral biases, such as riskaversion, loss-aversion, tendency to focus on short-term utility and so on.
As a concrete example, commuters may have some information on the expected congestion at each route via traffic reports or a cellphone app.However they also know that this information is inaccurate, and a risk-averse driver might take into account not just the expected congestion, but also the variance, standard deviation, a "safety margin" and so on.Further, different commuters may have different levels of riskaversion.
The implications of these limitations and biases on game playing can be similarly described: from the perspective of an outside observer (who cares about the actual total latency), the players are playing the "wrong game," either by applying some simple heuristics, or by optimizing a different utility function from the one in the game specification [Devetag and Warglien 2008].Moreover, different agents may have different perspective on the game, e.g.due to different levels of risk aversion.
Fortunately, as long as we can incorporate the biases that affect agents into suitably modified utility (or cost) functions, we can still use traditional equilibrium concepts.More formally, suppose that in the real underlying game G each agent i has some Permission to make digital or hard copies of part or all of this work for personal or classroom use is granted without fee provided that copies are not made or distributed for profit or commercial advantage and that copies bear this notice and the full citation on the first page.Copyrights for third-party components of this work must be honored.For all other uses, contact the owner/author(s).c YYYY Copyright held by the owner/author(s).1946-6227/YYYY/01-ARTA $15.00 DOI: 0000001.0000001utility function u i .Now, each player i sees her utility as some other function ûi , and thus we are interested in the equilibria of the biased game Ĝ comprised of modified utilities {û i }.Crucially, we still measure the quality of the outcome on the real game G (see Footnote 3 for further discussion).
Congestion games are a good testbed for such ideas.First, they have very convenient theoretical properties, such as the existence of a potential function and the uniqueness of equilibrium (up to identical utilities, in nonatomic games).Second, equilibrium inefficiency and in particular the Price of anarchy is very well understood.Third, several cognitive and behavioral biases have already been suggested and studied in the context of congestion games (see Related Work).
Our starting point in this paper is the smoothness framework in nonatomic congestion games in the Wardrop [1952] model.The smoothness framework connects a particular property of the edge cost functions (say, (1, 1 4 )-smoothness of affine functions), with a tight upper bound on the price of anarchy (PoA), this bound also being independent of the network topology [Roughgarden 2003]. 1  Smoothness for modified costs has also been considered; e.g. by Bonifaci et al. [2011] in the context of tolls that can also be viewed as a perturbation on utilities.The definition of smoothness naturally extends to account for bias, and the modified definition (which contains both the real cost c and the biased cost ĉ) guarantees a "biased" price of anarchy (BPoA) bound in a similar manner to standard smoothness PoA analysis.Moreover, it is easy to see that while [Bonifaci et al. 2011] considered a particular modified cost function, the same biased-smoothness approach works for any combination of c and ĉ.BPoA bounds have been similarly attained via smoothness for finite games with altruistic players [Chen et al. 2011].
Thus the PoA is not only robust to the network topology and to different notions of equilibrium [Roughgarden 2009], but also to players that are not playing exactly by the game specification.This observation brings about two important challenges: First, it is unlikely that all of the participants in a game have exactly the same bias (see example on risk-aversion).Thus we want to combine smoothness results of several different types to attain a single (biased) PoA bound.Second, can we find broad classes of games and behavioral biases for which smoothness applies?

Paper structure and Contribution
In Section 2 we briefly go over the standard model of nonatomic congestion games, price of anarchy and smoothness.In Section 3 we extend the definition of a smooth cost function to a more general notion of biased-smoothness, following similar extensions for specific modified costs [Bonifaci et al. 2011;Chen et al. 2011].We then provide positive results in both of the above directions: • For games over series-parallel networks, players are affected almost only by their own bias, as the negative externalities due to the biases of other agent types are bounded.• We conclude that under the same structural restriction, the BPoA in a game with a mixed population of different behavior types is bounded in terms of the biased smoothness parameters of all the participating types.We emphasize that this holds regardless of the specific biases, as long as for each agent type we have established smoothness (Section 4).
• In contrast, without such a structural restriction, even a small fraction of "bad" agents may inflict unbounded damage on the society.• We prove biased-smoothness for several classes of cost functions in non-atomic congestion games with particular behavioral biases: • We derive tight BPoA bounds for agents with tax sensitivity under polynomial cost functions of any degree d ≥ 1 (Section 5).• We improve the upper bounds of Meir and Parkes [2015] for pessimist agents (Section 6).• We significantly improve the upper bound of Nikolova and Stier-Moses [2015] for risk-averse agents, showing it is independent of the graph topology (Section 7).
We conclude with a brief comparison to related work, a discussion of our results, and suggest some directions for future work.

PRELIMINARIES
Nonatomic routing games.Following the definitions of Roughgarden [2003] and Roughgarden and Tardos [2004], a nonatomic routing game (NRG) is a tuple G = V, E, m, c, u, v, n , where , where (u i , v i ) are the source and target nodes of type i agents; • n ∈ R m + , where n i ∈ R + is the total mass of type i agents.The total mass of agents of all types is i≤m n i = n, and we assume unless specified otherwise that n = 1.
We denote by A i ⊆ 2 E the set of all directed paths between the pair of nodes (u i , v i ) in the graph.Thus A i is the the set of actions available to agents of type i.We denote by A = ∪ i A i the set of all directed source-target paths.We assume that the costs c e are non-decreasing, continuous, differentiable and semi-convex (i.e., xc e (x) is convex).Such cost functions are called standard [Roughgarden 2003].
Player-specific costs.A nonatomic routing game with player-specific costs (PNRG) is a tuple G = V, E, m, (c i ) i≤m , u, v, n .The difference from a NRG is that agents of each type i experience a cost of c i e (x) when x agents use edge e.We can have multiple types with the same source and target nodes to allow diversity of behavior.To avoid confusion, we refer to (v i , u i ) (or A i ) as the demand type and to c i as the cost type.Thus the type i specifies both the demand type and the cost type.
A PNRG is symmetric if all agents have the same demand type, i.e., A i = A for all i.A PNRG is a resource selection game (RSG) if (V, E) is a graph of parallel links.That is, if the action of every agent is to select a single edge.
Game states.A state (or action profile) of a PNRG is a vector s ∈ R |A|×m + , where s f,i is the amount of agents of type i that use path f ∈ A i .In a valid state, f ∈Ai s f,i = n i for all i.The total traffic on path f ∈ A is denoted by s f = m i=1 s f,i .Similarly, the total traffic on edge e ∈ E is denoted by s e = f :e∈f s f .
The social cost in a profile s in a NRG G is defined as ).An equivalent way to write the social cost is as This shows that the social cost only depends on the total traffic per edge.
Remark 2.1.This social cost provides an objective measure of the total latency in the system.However, the players in our model may act based on different cost functions, either because they have a different objective than simply minimizing their latency, or because they are affected by cognitive and behavioral biases.We provide examples of both types of discrepancies in Section 3.1.
We omit the argument G when it is clear from the context.We denote by OPT (G) some profile with minimal total cost, i.e.OPT (G) ∈ argmin s SC (G, s).
Equilibrium.A state s for an NRG is an equilibrium in game G if for every agent type i and actions That is, if no agent can switch to a path with a lower cost.This provides the analogy of a Nash equilibrium for nonatomic games, and it extends naturally for player-specific NRGs.
It is known that in any NRG there is at least one equilibrium, and that this can be reached by a simple best-response dynamic.Further, all equilibria have the same social cost and in every equilibrium all agents of type i experience the same cost [Aashtiani and Magnanti 1981;Milchtaich 2000;Roughgarden and Tardos 2004].Player-specific NRGs are also guaranteed to have at least one equilibrium [Schmeidler 1973], however, equilibrium costs may not be unique, and best-response dynamics may not converge.
Affine routing games.In an affine NRG, all cost functions take the form of a linear function.That is, c e (x) = a e x + b e for some constants a e ≥ 0, b e ≥ 0. The social cost can be written as SC (G, s) = e∈E a e (s e ) 2 + b e s e .Pigou's example is the special case of an affine RSG with two resources, where c 1 (x) = 1 and c 2 (x) = ax.We denote by G P (a, 1) the instance where c 2 (x) = ax (and c 1 (x) = 1), and G P (a, d) the instance where c 2 (x) = ax d (and c 1 (x) = 1).
The price of anarchy.Let EQ(G) be the set of equilibria in game G.The price of anarchy (PoA) of a game is the ratio between the social cost in the worst equilibrium in EQ(G) and the optimal social cost [Koutsoupias and Papadimitriou 1999].Since all equilibria have the same cost, we can write PoA(G) , where s * is any equilibrium of G.For example in affine NRGs, it is known that PoA(G) ≤ 4 3 , and this bound is attained by the Pigou example G P (1, 1) [Roughgarden and Tardos 2004].

BIASED COSTS
Given an NRG G = V, E, m, c, u, v, n and modified cost functions ĉi e for every type i ≤ m and edge e ∈ E, we obtain a PNRG Ĝ = V, E, m, (c i ) i≤m , u, v, n .We assume that players act based on their modified cost functions, irrespective of whether this is a rational behavior or not (see Remark 2.1).We refer to Ĝ as the biased game, where every agent of type i experiences a cost of ĉi e rather than c e .
Playing the Wrong Game A:5 Both games G and Ĝ have a role in our model.
where Ĝ is a biased version of G.
The way we interpret G is that players play the wrong game Ĝ (and thus, it is the equilibria of this game that matter), whereas their true costs are according to game G.In other words, the biased costs ĉi e affect behavior while the actual cost in state s is the original true cost c(s), and it is this true cost that determines welfare). 3iased Price of Anarchy.We measure the price of anarchy in a game with biased costs by comparing the equilibria of Ĝ to the optimum of G. Formally: . (3) In the uniform bias case where ĉi = ĉ for all i, the game Ĝ is just another NRG, and thus has a potential function.
In the general case, an equilibrium of Ĝ, and thus of G, always exists, although Ĝ may not be a potential game.Existence follows from general existence results on nonatomic games with convex strategy spaces and continuous utilities [Schmeidler 1973].

Biased Cost Functions
Uncertainty and Risk Aversion.A simple form of biased cost is induced by what we term pessimism, and can be thought of as a special case of risk-aversion.Suppose that whenever faced with some load s e on edge e, an agent believes this number to be inaccurate, and takes a safety margin by playing as if the actual load is r • s e (for some fixed private parameter r ≥ 1).
Such an agent will play as if every cost function c e is replaced with a new cost function ĉr e , where ĉr e (x) = c e (rx) (see Fig. 1(b)).This is equivalent to the worst-case-cost (WCC) model of Meir and Parkes [2015], which models agents with strict uncertainty over the actual costs.
A somewhat different model considers the uncertainty induced by an arbitrary noise distribution ǫ e (x) (with mean 0), by setting ĉγ f (s) = c f (s) + γvar( e∈f ǫ e (s e )), indicating that agents prefer paths with a lower variance [Nikolova and Stier-Moses 2015].Since the variance is additive along the path, we can rewrite the edge cost as ĉγ e (x) = c e (x) + γvar(ǫ e (x)).We assume that var e (x) is bounded by κc e (x) for some constant κ > 0 (see Fig. 1(d) for an example that violates this assumption).
The previous examples of biased games assumed players' perceptions that are biased or misguided.The next two examples are cases where the objectives of the players and the analyst/designer differ (see Remark 2.1).
Sensitivity to tax.Suppose that a central authority imposes an optimal tax of xc ′ e (x) on every edge (where c ′ e (x) is the derivative of c e ).If agents treat the monetary tax Figure 1: Fig. 1a illustrates a game G that is a quadratic variation of the Braess paradox [Frank 1981].The other subfigures present three biased versions of G.For example, the game Ĝb is played by pessimistic agents, with parameter r = 3. Hence the edges with fixed costs do not change, but the biased cost on the edge a-b for example is ĉu−a and the cost due to delay in the same way, then we get biased cost function ĉe (x) = c e (x) + xc ′ e (x) = c * e (x), with the effect that the incentives are perfectly aligned with those of the society, and the PoA is 1 [Beckmann et al. 1956].We emphasize that the designer in this case is only interested in the cost stemming from congestion, and disregards any monetary transfers.
However since the basic cost is in terms of delay, and the tax is in terms of money, different agents may have different money value for time, and hence different tax sensitivity [Cole et al. 2003;Karakostas and Kolliopoulos 2004;Fleischer et al. 2004;Fotakis et al. 2010].This suggests a biased cost function ĉβ (x) = c(x) + βxc ′ (x) (see Fig. 1(c)).As discussed in [Yang and Zhang 2008] these individual differences may be unobservable, and hence cannot be taken into account by the tax mechanism.
In our case, the designer does not know the value of β (or its distribution), and does not try to fit the tolls to the present population.Rather, we are interested in the effect of the standard, marginal-cost toll scheme (i.e., optimal for unbiased agents) on the social cost as β varies.
Capacitated resources.Suppose that each edge has a hard capacity limit L e , meaning that a state s is valid only if s e ≤ L e for all e ∈ E. We can translate this restriction to an endogenous bias, where ĉe (x) = c e (x) if x ≤ L e , and some large constant M > 0 otherwise.Since this would make ĉe discontinuous, we can define a similar function that only starts to increase after L e − δ, for example ĉδ e (x) = c e (x)(yM + 1)y 2 for all x ≥ L e δ, where y = x−(Le−δ) δ .This means that c δ e is standard, equals to c e in the range x ≤ L e , and is greater than M in the range x ≥ L e .As δ → 0, game Ĝ approaches the capacitated game.The BPoA then compares the cost of an equilibrium subject to capacity constraints to that of the optimal uncapacitated flow. 4ther types of biases that can be similarly modeled but are not studied in our paper are idiosyncratic preferences to certain strategies [Sandholm 2007], and altruistic behavior [Caragiannis et al. 2010;Chen et al. 2011].

Smoothness for Biased Costs
Our goal is to provide bounds on the biased Price of Anarchy for a given game with biased costs G, Ĝ .That each of c e and ĉi e are smooth is insufficient to provide such a bound.
Example 3.2.Consider a game with pessimistic agents G r = G r , Ĝr r where G r = G P (2/r, 1), i.e., the Pigou example.For any r the equilibrium of Ĝr r (and thus of G r ) is the same: 1/2 of all agents use each resource.However as r increases, the optimal solution of G shifts more agents to the second resource, and the optimal social cost decreases to 0. Thus the gap between the equilibrium cost and the optimal cost (the BPoA) goes to infinity with r.
We extend the definition of smoothness to games with biased costs in a way that takes into account both c and ĉ.This technique has been applied before for specific modified costs, for example nonatomic games with restricted taxes [Bonifaci et al. 2011] and atomic games with altruistic players [Chen et al. 2011].We provide an extension that handles general biases. (4) It is instructive to check the familiar case where there is no bias.Indeed, if ĉ = c, and c is (λ, µ)-smooth, then and Eq. ( 4) collapses to the standard definition of smoothness from Eq. ( 2).
Recall that the PoA of a (λ, µ)-smooth game is bounded by λ 1−µ .This bound extends to games with biased costs that are biased-smooth and when all agents have the same bias.
THEOREM 3.4 ([CORREA et al. 2008;ROUGHGARDEN 2009;BONIFACI et al. 2011]).Consider a game with biased costs G where every cost function c e is ( λ, μ)-biased smooth w.r.t.biased cost function ĉi e .Let s * be any equilibrium of the game Ĝ, and s ′ any valid state.Then SC (G, s * ) ≤ λ 1−μ SC (G, s ′ ).PROOF.The proof extends the standard proof of PoA bounds for nonatomic congestion games via smoothness arguments.In any equilibrium s * , the variational inequality e ĉe (s * e )s * e ≤ e ĉe (s * e )s ′ e holds (see [Correa et al. 2008;Roughgarden 2009]).Thus, Similarly, in the equilibrium s * b , s * d there is too little or too much traffic on the long path, respectively.An alternative derivation of optimal taxation.We can also check that the extension provides the right result in regard to modified costs c * (x) = c(x) + c ′ (x)x that represent optimal taxation and should lead to optimal play [Beckmann et al. 1956].Let's confirm this result via a biased-smoothness argument.
thereby affirming that the price of anarchy of G, G * is 1.

A:9
For example, the biased game Ĝc in Fig. 1 has optimal taxation, and we can see in Fig. 2 that its equilibrium x * c minimizes the social cost.
Robustness to small biases.Consider some bias s.t.

DIVERSE POPULATION
In general, there may be many types of agents in a game, each with a different behavioral bias.For example we can have agents with various levels of tax-sensitivity or pessimism, a small number of adversarial agents, and so on.Our main goal is to provide claims about the welfare achieved in the equilibria that correspond to such populations.
We define the following variations of a game G, given a set of biased cost functions (c i ) i≤m : • G i is obtained from G by setting the demand type of all agents to A i .Game G i is a symmetric NRG, where all costs are as in G.
by setting the cost type of all players to ĉi .Game Ĝi is a symmetric NRG.This corresponds to game with biased costs G i = G, Ĝi , where all agents have the same bias.
Also recall our previous definitions: • Ĝ is a PNRG that is obtained from G by setting the cost type of each type i agent to ĉi .This corresponds to game with biased costs G = G, Ĝ , where the modified cost function is obtained for each agent type via a distinct, type-specific mapping.
Suppose we can show biased smoothness of ( λi , μi ) for each of the uniform games with biased costs G i .The primary question is whether we can get a bound on BPoA(G) in terms of the smoothness parameters, ( λi , μi ) i≤m , for each type.Unfortunately, in the most general case of nonatomic congestion games even a small fraction of "bad" agents can arbitrarily increase the BPoA (see Section 4.4).
To obtain a positive result that remains general to different cost functions and behavioral biases, we restrict the structure of the network to series-parallel graphs.In Section 6, we show a bound for general networks, when we consider more restricted cost functions and biases.

Directed Series-Parallel Graphs
A directed series-parallel (DSP) graph is an acyclic directed graph (V, E, u, v) with a source u and a target v, and is composed recursively.
The basic graph is a single directed edge u − v. Two DSPs ) can be composed to obtain a new DSP either in a serial way: merge u 1 with v 2 ; or in a parallel way by merging u 1 with u 2 , and v 1 with v 2 .A (non-directed) series-parallel graph is obtained by taking a DSP and removing edge orientations.
A directed series-parallel game (DSPG) is a nonatomic congestion game over a DSP graph.RSGs are a special case of DSPGs in which all edges are composed in parallel.
We assume for simplicity that all cost functions in DSPGs are either strictly increasing or constant.In addition, and w.l.o.g., we assume there are no two paths with the same constant cost between any pair of nodes (otherwise we can treat them as a single path).
Let s i,e be the mass of type i agents on path f in state s.We have n i = f ∈A s i,f for any state s and type i.Denote the support of type i strategy by is the same, and weakly lower than the biased cost of any path f ′ / ∈ S i .We denote by SC i (G, s) = f ∈Si s i,f c f (s) the cost experienced by type i agents in state s.By summing over all types, we get the social cost: Our proof strategy is to first show a useful property of an equilibrium in seriesparallel graphs.Then we will proceed to bound the cost for each type (SC i (G, s)) separately and combine the bounds.
Let s * be an equilibrium of a DSPG with biased costs Ĝ, and s i be an equilibrium of the game Ĝi (i.e., the game in which the demand and cost types of all players are those of type i agents).
LEMMA 4.1.For any used path f ∈ S * i , s i f ≥ s * f .In particular, for any e ∈ S * i , s i e ≥ s * e .PROOF.The proof is by induction on the number of merging steps in the construction of the underlying graph.The base case for an edge is immediate.
If the DSP is composed of two smaller DSPs in series, then we can simply analyze each one as a separate game.An equilibrium of the full game projected over each subgame does not change, and thus we get the lemma.
For composition in parallel, the graph (V, E) is attained my merging two graphs (V 1 , E 1 ) and (V 2 , E 2 ) with the same source and target u, v. Denote the observed cost to type i agents in s * (regardless of their chosen path) by c * = ĉi (f, s * ).
By construction, there is no directed path in (V, E) that contains edges from both E 1 , E 2 .Denote by n * 1 , n * 2 the populations of agents in s * choosing paths in E 1 , E 2 , respectively (note that these are disjoint sets and 1 and likewise for (V 2 , E 2 ).Consider the equilibrium s i (in the complete game) when all agents are of type i.
1 contains some type i agents.Similarly partition the population into n i 1 , n i 2 according the their path choice in s i . 5Assume toward a contradiction that Since the total mass of agents in fixed, some other u i − v i paths must have more agents in s i .Crucially, s i remains the equilibrium of each subgame, that is, if we consider the agents of n i 1 playing on (V 1 , E 1 ) and likewise for the other subgame.
, then consider the game G 1 alone with |n * 1 | agents of type i.Denote by s (1) the attained equilibrium, which we now compare with s i (on the graph (V 1 , E 1 )).
Since we only decreased the number of agents, and in both games they are all of type i, we get an equilibrium s (1) f ≤ s i f for all f .In particular, s (1) However by our induction hypothesis applied to game G 1 , we should have s (1) f ≥ s * f for any path f ⊆ A i ∩ E 1 , and for f ′ in particular.This is a contradiction. Thus In particular there is some path We know that all agents in s i observe the same cost , whereas for any other path f (either in E 1 or E 2 ) we have ĉi (f, s i ) ≥ c i .Also ĉi (f ′ , s * ) = c * and the cost in s * is at least c * for any path including f ′′ .Therefore: This entails that (a) the costs of all edges along the paths f ′ , f ′′ are constants (as they are not strictly increasing); (b) that the paths f ′ , f ′′ have the same constant cost.This is in contradiction to our earlier assumption.

Fraction-independent bound
The first bound that we obtain is independent of the fractions of each type of agent in the population, and is useful when there is a large group of agents of the same type.PROPOSITION 4.2.Let s * be an equilibrium of DSPG Ĝ.For any type i, That is, if we only look at the cost to type i agents under the equilibrium s * of the game with biased costs, it is not worse than the overall cost in the equilibrium s i of the game Ĝi when all agents are of type i.In addition, the overall cost of this equilibrium is not much worse than the optimal social cost in this game Ĝi when all agents are of type i.
Prop. 4.2 has two immediate corollaries.First, the presence of a small fraction α of agents with arbitrary or adversarial behavior can harm the remaining agents by a factor of no more than 1/(1 − α): THEOREM 4.3.Suppose that a game with biased costs G = G, Ĝ is a symmetric game over a DSP network, where the total fraction of agents types other than i, j =i n j ≤ α, for some 0 < α < 1.Let s * be an equilibrium of G. Then there is some As before, λi and μi are two constants s.t. each c e is λi 1−μ i w.r.t.c i e .
PROOF.Let s be the optimal state where type i agents suffer the highest cots (since G is symmetric, we can swap agents arbitrarily without changing the social cost).Since G is symmetric, G i = G.Then by Prop.4.2, We also obtain a price of anarchy bound, by summing Eq. ( 5) over all types.This bound is useful when there are only a few types of agents.THEOREM 4.4.Suppose that G = G, Ĝ is a symmetric game over a DSP network.

Fraction-dependent bounds
The previous section provides a bound that only depends on the types in the population, and not in their relative frequency.However this bound may be too lax if there are many types, or when there is a small fraction of agents with high BPoA.
In this section we provide our main result w.r.t.diverse populations with general behavioral biases.
In the next proposition and theorem, each c e is (λ, µ)-smooth, each ĉi is (λ i , µ i ) smooth, and c e is ( λi , μi )-biased-smooth w.r.t.ĉi .Recall that ni n is the fraction of agents of type i.
PROPOSITION 4.5.Suppose that game with biased costs G = G, Ĝ is a DSPG with convex cost functions.Let s * be an equilibrium of G. Then for any type i, If the game is also symmetric, then OPT (G i )OPT (G) for all i.As an immediate corollary we get a bound on the BPoA of G.
THEOREM 4.6.Suppose that game with biased costs G = G, Ĝ is a symmetric DSPG with convex cost functions.Then BPoA(G) In other words, the BPoA of a diverse population on a DSP network is a weighted average of their worst-case PoAs (multiplied by a constant that does not depend on the network or the population structure).
Example 4.7.Consider a bias that maps affine costs to affine costs (e.g., tax sensitivity).We have that µ Similarly, for quadratic costs µ ACM Transactions on Economics and Computation, Vol.V, No. N, Article A, Publication date: January YYYY.

Playing the Wrong Game
A:13 Proof of the main result.Let G * be a game with biased costs derived from G, where the biased costs are defined as c * (x) = c(x) + c ′ (x)x.
This not true for non-convex functions, even if c(x)x is convex.For example, for c(x) = √ x and values close to 0, the derivative (and thus c * (x)) goes to infinity.For linear functions the bound is tight.Consider for example c(x) = x (for which λ = 1, µ = 1 4 ), then c * (x) = c(2x PROOF OF PROP.4.5.In equilibrium s * of Ĝ, all type i agents suffer the same biased cost F = ĉi (f, s * ) on all f ∈ S * i .In s i (the equilibrium of Ĝi ) the cost for all agents is F i = ĉi (f, s i ) where f is used path (in S i ).Take an arbitrary path f ′ ∈ S * i ∩ S i , then by Lemma 4.1, s * f ≤ s i f and thus F ≤ F i .We denote the optimal state by OP T (G i ) = (s o f ) f ∈A .Summing over the cost of all agents in s i , and using smoothness of Ĝi :6 (from Lemma 4.9) On the other hand, we lower bound the total experienced cost n i F in terms of the actual cost of s * .For any f ∈ S * i , by Lemma 4.8 Combining the above bounds, as required.

Necessity of restricted structure
Lemma 4.1 requires that the network is series-parallel.Without this, even a small fraction of biased agents can increase the cost for all others by an arbitrary factor.
PROPOSITION 4.10.Consider any graph (V, E) that is not directed series-parallel.Then for any ǫ > 0 and any M ′ > 0, there are symmetric games G, Ĝ with equilibria s * , ŝ * respectively, such that: • Only a fraction ǫ of the agents in Ĝ are biased.
• There are edges used in s * such that ŝ * e > s * e .

• For the unbiased type
PROOF.It is known that 2-terminal network is directed series-parallel, if and only if it contains the Braess graph (the graph that appears in Fig. 3) as a topological minor [Milchtaich 2006].Thus it is sufficient to construct a bad example on the Braess graph to prove the proposition.
Indeed, consider the network in Figure 3, where the value M is some large number that will be set later on.Suppose the total mass is one unit, and that all agents have need to flow from u to v.In the game G, costs are as in Figure 3 (a).Thus in the equilibrium s * , the agents will split evenly among the paths u-a-v and u-b-v, so that each agent incurs the same cost of (2 • 1/2) M = 1 M = 1, and thus SC 1 (s * , G) = 1 − ǫ.Now, denote the unbiased agents as type 1, and suppose that some fraction ǫ > 0 of agents is of a type 2, with biased costs ĉ2 .See Figure 3 (b).
In the equilibrium ŝ * , all type 2 agents will select the path u-a-b-v, whereas all type 1 agents will split evenly.Thus there will be ŝ * e = 1 2 + ǫ 2 > s * e agents on each edge s-u and b-v.Note that this already shows that Lemma 4.1 does not apply for our game.

Playing the Wrong Game
A:15 We now compute the cost for the type 1 agents in ŝ * .Set M > M ′ ǫ .For every such agent taking the path s − a − t, the cost is Thus the total cost for type 1 agents is Since the cost function used in this proof are not smooth, this does not violate the BPoA bound in Theorem 4.6.See Proposition A.1 in the appendix for another example with smooth costs and unbounded BPoA.

BIASED SMOOTHNESS FOR TAX-SENSITIVE AGENTS
In this section we consider the role of behavioral bias in the context of tax payments.In particular, suppose that the center imposes the optimal tax of c ′ e (x)x on every edge e.For an agent with tax sensitivity β ≥ 0, we have ĉβ (x) = c(x) + βc ′ (x)x, and the monetary part of the cost is adjusted by a factor of β.We will assume that all agents have the same bias, and provide bounds on BPoA(G, β) ≡ BPoA(G, Ĝβ ).That is, to bound the efficiency loss in games where all agents have a bias β.When agents differ in their bias the results can be combined with the analysis of bounds under diverse populations.
We consider four classes of games by generality of the cost functions: all standard cost functions, convex functions, polynomial functions with positive coefficients, and affine functions.For example, G affine stands for an arbitrary game with affine costs.Unlike in the previous section, all of our BPoA bounds hold regardless of the network topology.The results are summarized in Tables I and II.
The BPoA bound follows immediately from Cor. 3.5.

Affine costs
We can derive tighter smoothness bounds for more specific classes of cost functions and biases.The results for affine cost functions are summarized in Table II and in Figure 4.

Playing the Wrong Game
Table I: Biased-smoothness bounds for tax-sensitivity Table II:  For reference, we also show the upper bounds derived for arbitrary cost functions (in light gray, from Props.5.1 and 5.2); using convexity only (thick green line, from Prop.5.3); and from Meir and Parkes [2015] (dashed).All bounds apply to pessimistic and risk-averse agents (see Section 6).
While the bounds for affine functions follow as a special case from Prop.5.6 and Prop.5.7, we lay out the (shorter) proofs for intuition.We need the following inequality, which is often used in the analysis of nonatomic games: PROOF.We need to show that c(x We have: We get the upper bound on the BPoA from biased-smoothness and Corollary 3.5.The lower bound is attained by the Pigou example G P (a, 1), for a = 1 β+1 .This follows from the analysis in Meir and Parkes [2015], as we explain in the next section, but we directly derive the lower bound for completeness in the appendix.
The lower bound is attained by the Pigou example G P (a, 1), for a = 2β (β+1) 2 .Higher order polynomials PROPOSITION 5.6.For tax sensitivity β ≥ 1, polynomial cost functions of degree  II).For reference, we show the upper bounds derived for arbitrary cost functions (in light gray, from Props.5.1 and 5.2); and using convexity only (thick green line, from Prop.5.3).

This is a tight bound.
Note that for β = 0, we get the familiar PoA bound of , whereas for β = 1 we get that the BPoA is 1.PROPOSITION 5.7.For tax sensitivity β ≥ 1, polynomial cost functions of degree d are . This is a tight bound.Note that as degree d grows, the bound approaches 1 − 1 e β .In contrast, by using Prop.5.3 we only get a bound of µβ + O(1), which tends to β + O(1) as the degree d grows.That is, the bound attainable by only using the convexity property of polynomial costs is greater by a multiplicative factor.
In particular, for both propositions we get tight bounds for the BPoA under quadratic cost functions, which are presented in Table II and in Fig. 5.

BIASED SMOOTHNESS FOR THE WCC MODEL
Suppose now that agents are pessimistic, in the sense that they play according to a congestion amount that is larger by a factor of r > 1 than the true congestion [Meir and Parkes 2015], that is, ĉr (x) = c(rx).7For affine cost functions, it holds that and pessimism with factor r ≥ 1 coincides with tax-sensitivity of β = r − 1.
As a result, we can immediately apply the upper bounds for tax-sensitive agents to pessimistic agents with affine cost functions (see Fig. 4).
Lower bounds for pessimistic agents were previously derived in Meir and Parkes [2015], and thus we can also go in the other direction and apply them to the taxsensitive agents.This correspondence between tax-sensitive agents (for affine costs) and risk-averse agents provides an explanation as to why a little uncertainty (or pessimism) initially improves the BPoA, but too much of it hurts: the biased cost increases (due to pessimism) until it reaches the point of optimal taxation, but then goes past it.
Diverse populations.Returning to diverse populations, another result from Meir and Parkes [2015] that we can apply to tax-sensitive agents with affine costs, relates to diverse populations.Consider a game with heterogeneous tax sensitivities (β i ) i≤m , and let β = min i β i , and β = max i β i .THEOREM 6.1 (DIRECT COROLLARY FROM MEIR AND PARKES [2015]).For any affine NRG G and tax sensitivities β = (β i ) i≤m : We conjecture that a similar technique can be used to prove similar results for any game with a diverse population of tax-sensitive agents, and leave this for future work.Theorem 4.6 and Theorem 6.1 provide us with two different tools to derive BPoA bounds for diverse populations: the former applies only for DSPGs but takes agents' relative quantities into account, whereas the latter provides a bound in terms of the worst type, but applies for any NRG.
Higher order polynomials.We can apply a similar analysis to other cost functions, although this may lead to different results.Just considering higher order polynomials breaks the correspondence with tax-sensitivity.For example, for quadratic costs: In particular, the BPoA is no longer linear in r.For example, for quadratic cost functions with r > 2, the BPoA is between r 2 8 and r 2 4 (see appendix).More generally for polynomials of degree d, the BPoA is r Θ(d) .In addition, while for some value of r > 1 the social cost improves, it does not reach a BPoA of 1, except for affine cost functions.
Nikolova and Stier-Moses bounded the "Price of Risk Aversion" (PRA), which is the ratio between the social welfare in the biased equilibrium and that in the non-biased equilibrium.Their main result is that the PRA is upper bounded by 1 + κγη, where η is a parameter that depends on the network and may be as large as the number of vertices.In particular this leads to a bound of BPoA(G µ,η , γ, κ) ≤ (1 + γκη) 1 1−µ for any (1, µ)-smooth game with network parameter η under risk aversion with parameters γ, κ.

Playing the Wrong Game
A:21 PROOF.Suppose first that x ′ ≤ x.Then (from smoothness of c) Next, suppose that x ′ > x.Then as required.
An immediate corollary is that for any (1, µ)-smooth game G µ under risk aversion with parameters γ, κ, we have This result improves not just the BPoA bound that follows from Nikolova and Stier-Moses [2015], but also their PRA bound.
Because the unbiased equilibrium is at least at costly as the optimal state, it follows that the PRA is at most the (biased) PoA, and is thus upper bounded by (1 + γκ) 1 1−µ .Crucially, µ is fixed and does not depend on the network structure.This shows that the factor η is not so crucial when considering either the BPoA or the PRA.
We should note however that for some parameter values (namely when η is small and the actual equilibrium of Ĝ is not much better than the worst-case bound), the bound given by Nikolova and Stier-Moses [2015] is better than ours by a constant factor.

Related Work
Various modified cost functions have been considered in the literature of congestion games.A primary example is when taxes (or tolls) are imposed on players using an edge, where the travel time plus the imposed tax can be thought of as a modified cost function.Most related to our work, Bonifaci et al. [2011] design a particular tax mechanism that operates under various restrictions (e.g., taxes may not be too high), and use a smoothness argument to prove BPoA bounds in the resulting game.In a followup work, Jelinek et al. [2014] design optimal taxes for RSGs with restricted taxes, where agents have diverse tax sensitivities.While most work on taxes including the papers above focused on the design of good tax schemes [Cole et al. 2003;Fleischer et al. 2004], we analyze the affect of a given (standard) taxation scheme on the price of anarchy.Chen et al. [2011] applied smoothness analysis to provide BPoA bounds for various games (including atomic congestion games) where players are altruistic, i.e., part of their utility is derived from the social welfare.They also bounded the BPoA in games with diverse levels of altruism via direct analysis.We note that in the nonatomic model, altruism amounts to averaging the latency with a number that is constant over all actions, and thus does not affect players' behavior or the BPoA.
[ Acemoglu et al. 2016] study nonatomic congestion games where some agents are unaware of the existence of certain edges, which is equivalent to having a "wrong" cost function that assigns infinite costs to some edges.They prove that in DSPGs such ignorance can only lead to a worse equilibrium than under true information, yet worstcase PoA does not exceed the bound of λ 1−µ .Christodoulou et al. [2014] proposed coordination mechanisms for nonatomic congestion games, where the modified cost can be increased arbitrarily.However in contrast with most of the literature on introducing tolls into congestion games, the modified costs are considered "real" rather than "biased," in the sense that the social cost consists of both the latency and the tolls.
Another behavioral bias that has been recently studied in congestion games is risk aversion [Ashlagi et al. 2009;Angelidakis et al. 2013;Piliouras et al. 2013;Nikolova and Stier-Moses 2015;Meir and Parkes 2015].In some of these models, the behavior of a risk averse agent coincides with that of a rational agent with a particular biased cost function (see Sections 6, 7).We make a detailed comparison in the relevant sections.
Finally, Babaioff et al. [2007] consider congestion games where some of the players are malicious.We can reformulate such a game as a biased game, where malicious behavior corresponds to an arbitrary bias.

Conclusion
We have considered strategic settings in which participants are playing the wrong game, and perceiving utilities in some inaccurate or biased form.Whether these modified utilities, and thus deviations from rational play, come from a cognitive limitation, a behavioral bias, or a different perception in regard to taxes or other payments, it is important to understand how the equilibria of the game are affected.
Biased smoothness is a tool that enables PoA analysis under such modified utilities.Our work is the first to provide PoA bounds on populations with arbitrary biases, diverse across agents, in nonatomic congestion games.The analysis framework reduces the problem to that of analyzing the smoothness parameters of each payoff and behavioral agent type.
Future research directions include deriving PoA bounds under various other types of biases and for more classes of cost functions.A more fundamental question is whether our results on diverse populations with either specific or arbitrary biases can be extended, namely relaxing some of the assumptions in our main results-in particular the restrictions on the graph structure.The game G with its base cost functions.There are q disjoint paths of length 3 from s to t.
PROOF.Consider the network in Figure 6, where the number of parallel paths q is set s.t.q > 2M ǫ , d > max{log q + 1, 10}, and a = (q/2) d .The optimal flow is when agents evenly split among the q paths: then the cost to every agent is This is also the social cost.If all agents were of type 1, then the optimal flow would also be the unique equilibrium.Now consider the behavior of the type 2 agents.The modified cost on the source/target edges becomes ĉ2 (x) = a(rx) d = a8 d x d .The type 1 agents (of mass at least 0.5 still split evenly, thus x > 1/2q, and So for the type 2 agents, the source/target edges (e.g s-a1) become so expensive, they all must choose the path s-s'-a1-a2-b1-b2-...-q1-q2-t'-t.Therefore, in equilibrium s * we have on each middle edge 1−ǫ q + ǫ > ǫ agents.The (real) cost to each agent is thus at least ǫ, which means SC (G, s * ) ≥ ǫ.
Joining the above results, Let s * = s * 2 be the equilibrium load on edge 2. In equilibrium, the biased cost on both edges is the same, thus 1 Proposition 5.5.For tax sensitivity β ≥ 1, affine cost functions are 4β .This bound is tight.
The equilibrium cost is thus: Consider the state s ′ where s ′ = 1.Then SC (s ′ ) = a.The BPoA is thus, .This is a tight bound.

Figure 2 :
Figure 2: The social cost in game G from Fig. 1 as a function of the traffic on the edge u − a (assuming symmetric flow).The equilibria of all four games are presented.
e (s ′ e )s ′ e + μc e (s * e )s * e ] = λSC (G, s ′ ) + μSC (G, s * ).(by Def.3.3) We get the bound by reorganizing the terms.The only part that differs from the standard smoothness is the first inequality.COROLLARY 3.5.Consider a game G where each cost function c e is ( λ, μ)-biased smooth w.r.t.ĉe .Then BPoA(G, Ĝ) ≤ λ 1−μ .The equilibrium of each of the biased games from Fig. 1 are presented in Fig. 2. The equilibrium of the real game s * a is suboptimal, as all agents take the long path

Figure 3 :
Figure 3: All agents need to select an u-v path.Some fraction of type 2 agents adopt the modified cost function ĉ2 , where M ≫ 1 is some large constant.

Figure 4 :
Figure 4:The double red lines are the tight bounds on the PoA in NRGs with affine costs, for agents with tax-sensitivity of β ≥ 0 (see Props.5.4 and 5.5).For reference, we also show the upper bounds derived for arbitrary cost functions (in light gray, from Props.5.1 and 5.2); using convexity only (thick green line, from Prop.5.3); and fromMeir and Parkes [2015] (dashed).All bounds apply to pessimistic and risk-averse agents (see Section 6).

−Figure 5 :
Figure 5: The double red lines are upper and lower bounds on the BPoA in NRGs with quadratic costs, for agents with tax-sensitivity of β ≥ 0 (see TableII).For reference, we show the upper bounds derived for arbitrary cost functions (in light gray, from Props.5.1 and 5.2); and using convexity only (thick green line, from Prop.5.3).
Figure 6:The game G with its base cost functions.There are q disjoint paths of length 3 from s to t.