# Shortest Paths from a Group Perspective—A Note on Selfish Routing Games with Cognitive Agents

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction

- Varying probabilities of agents acting in a non-selfish way.
- Varying levels of uncertainty in travel time information. That is, the information available on network congestion-status may be fuzzy for some agents.

- (a)
- Real-time traffic applications (like Waze) are dependent on the number of users collecting & providing data. Such real-time applications have a high market share in certain countries, but they do not have a significant market penetration in all regions of the earth. In some European capital cities the numbers of Waze users are at a maximum. For example, in Paris approximately 51,000 users per/1 million citizens use the app. Other cities lag behind, for example Vienna involves only 1000 users/1 million citizens [9].
- (b)
- Traffic Message Channels (TMC) provide traffic information for vehicle drivers sent via FM radio frequency. This information can be included into any satnav system for routing purposes. The system is designed so that traffic information is only provided for major traffic junctions/locations collected in a location table—which results in inaccuracies in the location and extent of any traffic congestion.
- (c)
- As TMC or FM radio provided traffic information requires that traffic information be collected, checked and published thereafter, there is a temporal delay between the incident and the publishing of the traffic information. In FM radio stations, the updates on traffic may be broadcasted every 30 min, which means an incident may have cleared by the time that the information is broadcasted.
- (d)
- As we deal with a dynamic situation, the traffic conditions ahead of an agent may alter as the agent moves toward an incident location that is along the desired route. Hence, any agent has to rely on a prediction on how the situation might change—especially when the agent does not receive any timely update on the situation as he/she comes closer to the incident location.

## 2. Network Flows, Routing Games and Selfish Routing

#### 2.1. Shortest Paths and Network Flows

_{e}:R

^{+}→R

^{+}. Such travel time functions are usually expressed as a function of traffic flow, and increases with increasing traffic. In most networks, a number of different directed paths exist that connect an origin or source node with a destination or node.

^{+}such that 0 ≤ f(u,v) ≤ c(u,v) ∀ (u,v) ∈ E (Capacity Constraints) and Σ

_{v}∈ V f(u,v) = Σ

_{v}∈ V f(v,u), for all u ≠ s,t (Flow Conservation constraints), where s, t denote the source and sink nodes of the network. Two of the most well-known flow problems are the max flow and the minimum cost flow problems [16].

#### 2.2. Routing Games and Selfish Routing

## 3. Quantifying the Impact of Cognitive Agents within a Collective of Players

_{e}:R

^{+}→R

^{+}. That is, we can express travel delay or latency as c

_{e}(x) = ax + b. A Graph G has k source and destination vertex pairs {s

_{1},t

_{1}}, … ,{s

_{k},t

_{k}}. A simple pair of source and destination, s

_{i}− t

_{i}, is denoted as P

_{i}and the set of pairs is designated as P = {P

_{i}}. Any network flow is defined as a function f:P→R

^{+}, and a fixed flow f is defined as ${f}_{e}={\displaystyle \sum}_{P:e\u03f5{P}_{i}}{f}_{P}$. In addition, a finite, positive rate r

_{i}is associated with each pair (s

_{i},t

_{i}), which represents the demand for travel or flow between source s

_{i}and destination t

_{i}. Generally, a flow is feasible if $\forall i{\displaystyle \sum}_{P\in {P}_{i}}{f}_{p}={r}_{i}$. Each edge e ∈ E is given a load-dependent latency or travel time function that is denoted as l

_{e}(∙). The latency function is non-negative, differentiable and non-decreasing. Hence, the triple (G,r,l) represents a specific problem instance. The latency of a path P with respect to a feasible flow f is the sum of latencies of the edges in the path represented by ${l}_{P}\left(f\right)={\displaystyle \sum}_{e\in P}{l}_{e}\left({f}_{e}\right)$. The cost C(f) of an entire set of flows f in G is the total latency incurred by f and is defined by:

_{AB}= l

_{CD}= x/100 and the others are assigned l

_{BD}= l

_{AC}= 45—which are the latency functions used by Braess [8]. In order to evaluate the effects of cognitive agents within the context of Braess Paradox, an “additional” edge is depicted with a dashed line from node B to node C. The extra edge is assigned a latency function of l

_{BC}= 1.

_{BC}the players in the game will behave as players in a non-cooperative game. Hence, 2000 agents will take the path P

_{1}= {A,B,D} and the remaining 2000 agents choose path P

_{2}= {A,C,D} (see the left hand side of Figure 2). The given result is a flow at Nash equilibrium [25,26,27,28], which indicates that each agent is behaving “greedily”, without regard to the overall cost of travel on the network. Hence, each player travels along the minimum latency path currently available, with respect to the flow created by the other players. If a flow is at Nash equilibrium for an instance (G,r,l) assuming $i\in \left\{i,\dots ,k\right\}$ and ${P}_{1},{P}_{2}\in {P}_{i}$ with ${f}_{{P}_{1}}>0,{l}_{{P}_{1}}\left(f\right)\le {l}_{{P}_{2}}\left(f\right)$ then all used s

_{i}− t

_{i}paths have equal total latency. In the example employed here, the overall latency (cost of flow) C(f) equals 260,000 units (or 65 units of delay per agent). If the additional edge with high capacity (i.e., low latency l

_{BC}= 1) is added to the network, a flow at Nash equilibrium exists (assuming a non-cooperative game). The flow results in the following situation: All 4000 agents take the path P

_{3}= {A,B,C,D} (see the right hand side of Figure 2, indicated by the red colored edges). The unique flow at equilibrium has a total cost C(f) which equals 324,000 units (or 81 units of delay per agent) [7,13,26,29]. Therefore, adding an additional edge and associated capacity can actually impede traffic flow rather than improve traffic flow, given that the agents act in a non-cooperative manner. This is an instance of Braess paradox.

#### Methodology to Evaluate the Effects of Cognitive Agents

_{NS}. The values of the parameter can range from 0 to 100, where 0 indicates that all decisions taken are selfish and 100 assumes that all decisions taken are of non-selfish nature. Usually probability levels have values from 0 to 1. In this paper, we use probability values multiplied by the factor 100, for the sake of readability. A P

_{NS}value of 50 means that there is a 50% probability that the decision of an agent will be non-selfish. Thus, the agents in the simulation may have selfish and non-selfish behavior. A non-selfish decision means, that the agent will not take the “obvious” faster path P

_{3}= {A,B,C,D}, but chooses one of the “slower” paths P

_{1}or P

_{2}, for whatever reasons. In order to evaluate the effect of non-selfish behavior, we have varied an agent’s probabilities of non-selfish decisions, from 0 to 100 in 5-unit steps—i.e., 0, 5, 10, …, 100.

_{t}, to the path P

_{3}= {A,B,C,D} in the network. Hence, the total latency with uncertainty at P

_{3}is denoted as C

^{0}(f). The calculation of C

^{0}(f) is defined in Equation (2),

_{t}]. The ∆

_{t}is assigned a value ranging from 0 to 100 in 10-unit steps—i.e., 0, 10, 20, …, 100, according to the level of uncertainty that is applied in a specific test simulation.

_{NS}and uncertainty applied to latency of P

_{3}, 5000 simulation runs were performed (see Figure 3). In every simulation run 4000 agents have to travel from node A to node D, where they have to make decisions on the route taken based upon their cognitive and decision abilities. Overall, there are 231 combinations of P

_{(NS)}and ∆

_{t}. Given 5000 simulations for each distinct combination, there are 1,155,000 simulation runs. For each test run, we collected the following result variables: Total latency C

^{0}(f), the number of agents traversing edge e

_{AB}, the number of agents traversing edge e

_{BD}, the number of agents traversing edge e

_{AC}, the number of agents traversing edge e

_{CD}, the number of agents traversing edge e

_{BC}, and the number of selfish and non-selfish decisions made. For each distinct combination of ∆

_{t}and P

_{(NS)}the variables collected in each of the 5000 simulation runs were statistically analyzed. Hence, the mean value, the standard deviation and variance of each result variable for each combination of ∆

_{t}and P

_{NS}was calculated.

## 4. Experimental Results

^{∗1}) denotes that the additional edge BC is used, and superscript of (

^{∗2}) denotes that the edge BC is not used. This means that $\overline{{C}^{\prime}}{\left(f\right)}_{\Delta 0,NS100}$, marked with (

^{∗2}), denotes C(f) of the flow at Nash equilibrium without edge BC.

_{e}(f) of the Nash flow of the extended network—i.e., with edge BC. Based on these “anchors”, the variable conditions of uncertainty of travel times and variable probability levels of non-selfishness were tested. The calculation of C(f) and C

_{e}(f) is done according to the methodology mentioned in Section 3.1. Therefore, a non-cooperative game is created and evaluated until no agent can improve their individual situation by changing their behavior. Hence, C(f) results in 65 latency units per agent traveling from A to D, where 2000 agents traverse the edges AB-BD and the other 2000 agents choose AC-CD. For the network with the extra edge BC (having low latency) C

_{e}(f) results in 81 units of latency per agent. In this case all 4000 agents traverse the edges AB-BC-CD. This paradox, of higher latency values due to an extra high capacity edge, is described in literature as Braess Paradox (e.g., Reference [8]).

_{(NS)}). This is depicted in Figure 4 and Table 1. Generally, the prior statement holds true except for the set of latency times highlighted in orange in Table 2. The highlighted $\overline{{C}^{\prime}}\left(f\right)$ values at a given P

_{(NS)}level are the lowest calculated values for given ∆t values. With increasing ∆t the values of $\overline{{C}^{\prime}}\left(f\right)$ increase. In Figure 5 the behavior of latency values for varying P

_{(NS)}with a given ∆t value is depicted (see Table 1 for numerical values). There, the latency values for ∆t values 0, 30, 60, 100 are depicted, showing decreasing latency values per agent with increasing P

_{(NS)}. This monotonically decreasing behavior is present from P

_{(NS)}levels 0 to 90 (for ∆t ranging from 0–30), for P

_{(NS)}levels 0 to 85 (for ∆t ranging from 40–70), and for P

_{(NS)}levels 0 to 80 (for ∆t ranging from 80–100).

_{(NS)}values from 50 to 65 across all ∆t levels. In contrast to those numbers, the coefficient of variation for edges BD and AC are in the range between 8% and 395%, having decreasing coefficients of variation with higher P

_{(NS)}levels—except for P

_{(NS)}= 0. Hence, within the conducted test runs the standard deviation of edges BD and AC show higher values in comparison to AB and CD especially at low P

_{(NS)}and ∆t values. This is due to the fact that at low P

_{(NS)}and ∆t values edges BD and AC are not traversed by many agents, as most follow the path P

_{3}= {A,B,C,D}. The coefficient of variation for edge BC ranges between 0% and 401% showing a high influence of P

_{(NS)}levels—i.e., increasing P

_{(NS)}leads to increasing variation.

_{(NS)}) which are volatile. Volatility in this context indicates test runs with high standard deviations, which in turn are unstable in terms of the number of traversing agents. Hence, a forecast or simulation of such situations is hardly possible, due to the variability of the system itself. For the experimental settings in this paper the edges AB and CD have an average coefficient of variation of 16% which is lower than the average coefficient of variation for edges BD and AC (53%). Hence, we can assume that the number of traversing agents of AB and CD are considered more stable than on AC and BD. For low ∆t values and low P

_{(NS)}levels coefficient of variation for edges AC and BD show especially high volatility due to the fact that the number of agents traversing these edges is low. The edge BC also shows unstable behavior in the test runs where the path P

_{3}is seldom traversed.

_{(NS)}on the average travel time per agent is also worth evaluating. In general, both variables have an influence on $\overline{{C}^{\prime}}\left(f\right)$ per agent, while P

_{(NS)}has a greater impact on $\overline{{C}^{\prime}}\left(f\right)$ per agent, than ∆t. This is justified by the numerical values given in Table 1, and by the correlation coefficients given in Table 3 and Table 4. In the tables the dependence of the latency values on the variables ∆t and P

_{(NS)}is given, where Table 4 indicates that P

_{(NS)}has higher impact on the $\overline{{C}^{\prime}}\left(f\right)$ values due to higher correlation coefficients in comparison to Table 3.

_{3}the travel latency per agent is actually lower than in the network without the extra edge (see Figure 4 and Figure 5 and Table 1 and Table 2).

## 5. Conclusions and Future Work

#### 5.1. Crititcal Discussion of Obtained Results

#### 5.2. Future Work and Connection to ITS

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## Appendix A. Supplementary Results

Agents Traversing Edge AB and CD | |||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|

Travel Time Uncertainty ∆t | |||||||||||

P_{(NS)} | 0 | 10 | 20 | 30 | 40 | 50 | 60 | 70 | 80 | 90 | 100 |

0 | 4000 | 3974.81 | 3849.15 | 3734.76 | 3632.05 | 3547.62 | 3469.37 | 3401.86 | 3339.55 | 3295.05 | 3241.06 |

5 | 3900.07 | 3879.95 | 3758.69 | 3641.37 | 3547.75 | 3461.11 | 3384.27 | 3321.4 | 3257.16 | 3200.6 | 3162.78 |

10 | 3811.04 | 3774.2 | 3663.04 | 3555.4 | 3460.76 | 3379.4 | 3300.79 | 3242.36 | 3167.76 | 3120.97 | 3073.99 |

15 | 3708.08 | 3679.9 | 3580.8 | 3461.63 | 3376.74 | 3287.77 | 3215.32 | 3155.59 | 3101.99 | 3033.97 | 2986.3 |

20 | 3604.07 | 3583.16 | 3481.59 | 3381.52 | 3298.92 | 3199.16 | 3132.09 | 3080.07 | 3010.67 | 2959.26 | 2923.24 |

25 | 3505.82 | 3482.06 | 3386.4 | 3286.22 | 3196.58 | 3120.97 | 3048.38 | 2983.14 | 2933.44 | 2886.59 | 2838.81 |

30 | 3416.31 | 3384.71 | 3280.88 | 3199.04 | 3115.95 | 3046.93 | 2969.3 | 2922.98 | 2861.46 | 2809.34 | 2767.17 |

35 | 3294.03 | 3280.85 | 3192.62 | 3107.43 | 3042.72 | 2952.77 | 2894.61 | 2841.55 | 2794.96 | 2743.55 | 2707.56 |

40 | 3201.92 | 3176.84 | 3097.61 | 3025.9 | 2952.37 | 2879.66 | 2823.73 | 2775.24 | 2720.1 | 2675.26 | 2636.41 |

45 | 3089.87 | 3088.58 | 3009.95 | 2943.63 | 2872.17 | 2815.76 | 2748.1 | 2697.71 | 2652.49 | 2612.29 | 2569.12 |

50 | 2995.9 | 2994.26 | 2912.51 | 2852.01 | 2791.12 | 2731.5 | 2671.15 | 2629.46 | 2587.19 | 2548.21 | 2507.01 |

55 | 2898 | 2892.6 | 2827.96 | 2769.3 | 2724.01 | 2660.53 | 2608.99 | 2570.34 | 2526.3 | 2485.34 | 2460.01 |

60 | 2800.92 | 2798.16 | 2743.45 | 2702.34 | 2646.85 | 2587.73 | 2548.51 | 2503.55 | 2465.33 | 2432.68 | 2401.98 |

65 | 2694.2 | 2694.99 | 2658.81 | 2615.55 | 2568.78 | 2519.65 | 2478.97 | 2436.63 | 2404.11 | 2375.98 | 2348.21 |

70 | 2594.52 | 2596.72 | 2566.77 | 2525.5 | 2482.91 | 2443.28 | 2405.1 | 2371.54 | 2340.83 | 2315.53 | 2291.25 |

75 | 2500.25 | 2497.14 | 2469.29 | 2441.55 | 2402.34 | 2366.42 | 2336.04 | 2310.01 | 2282.15 | 2260.36 | 2241.7 |

80 | 2401.09 | 2394.62 | 2381 | 2353.58 | 2323.59 | 2292.75 | 2273.31 | 2250.37 | 2222.16 | 2205.35 | 2191.58 |

85 | 2296.75 | 2295.48 | 2290.35 | 2267.5 | 2242.06 | 2224.12 | 2205.9 | 2184.36 | 2166.27 | 2153.98 | 2141.78 |

90 | 2196.33 | 2202.09 | 2194.23 | 2174.59 | 2158.42 | 2150.37 | 2131.68 | 2118.38 | 2111.17 | 2100.74 | 2091.57 |

95 | 2094.3 | 2104.05 | 2092.21 | 2086.44 | 2081.78 | 2072.18 | 2065.88 | 2059.27 | 2054.71 | 2048.08 | 2043.54 |

100 | 2000 | 2000 | 2000 | 2000 | 2000 | 2000 | 2000 | 2000 | 2000 | 2000 | 2000 |

Agents Traversing Edge BD and AC | |||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|

Travel Time Uncertainty ∆t | |||||||||||

P_{(NS)} | 0 | 10 | 20 | 30 | 40 | 50 | 60 | 70 | 80 | 90 | 100 |

0 | 0 | 25.2 | 150.85 | 265.25 | 367.96 | 452.39 | 530.65 | 598.17 | 660.47 | 705 | 758.97 |

5 | 99.93 | 120.06 | 241.32 | 358.65 | 452.26 | 538.91 | 615.76 | 678.64 | 742.88 | 799.45 | 837.27 |

10 | 188.96 | 225.81 | 336.97 | 444.62 | 539.27 | 620.64 | 699.24 | 757.69 | 832.29 | 879.08 | 926.07 |

15 | 291.92 | 320.09 | 419.22 | 538.4 | 623.3 | 712.27 | 784.74 | 844.47 | 898.08 | 966.08 | 1013.76 |

20 | 395.93 | 416.84 | 518.42 | 618.52 | 701.13 | 800.9 | 867.97 | 920 | 989.38 | 1040.81 | 1076.82 |

25 | 494.18 | 517.94 | 613.63 | 713.82 | 803.47 | 879.08 | 951.68 | 1016.92 | 1066.64 | 1113.47 | 1161.24 |

30 | 583.68 | 615.29 | 719.14 | 800.99 | 884.09 | 953.13 | 1030.75 | 1077.09 | 1138.6 | 1190.71 | 1232.91 |

35 | 705.97 | 719.16 | 807.4 | 892.6 | 957.32 | 1047.28 | 1105.45 | 1158.51 | 1205.12 | 1256.52 | 1292.52 |

40 | 798.08 | 823.16 | 902.42 | 974.12 | 1047.68 | 1120.38 | 1176.34 | 1224.83 | 1279.97 | 1324.81 | 1363.66 |

45 | 910.13 | 911.42 | 990.07 | 1056.4 | 1127.87 | 1184.3 | 1251.96 | 1302.35 | 1347.58 | 1387.78 | 1430.94 |

50 | 1004.09 | 1005.74 | 1087.51 | 1148.02 | 1208.94 | 1268.55 | 1328.9 | 1370.62 | 1412.87 | 1451.87 | 1493.05 |

55 | 1102 | 1107.41 | 1172.06 | 1230.75 | 1276.05 | 1339.54 | 1391.08 | 1429.74 | 1473.78 | 1514.74 | 1540.07 |

60 | 1199.08 | 1201.85 | 1256.56 | 1297.71 | 1353.2 | 1412.33 | 1451.55 | 1496.53 | 1534.73 | 1567.4 | 1598.09 |

65 | 1305.8 | 1305.01 | 1341.21 | 1384.49 | 1431.27 | 1480.41 | 1521.1 | 1563.45 | 1595.96 | 1624.09 | 1651.85 |

70 | 1405.47 | 1403.29 | 1433.25 | 1474.53 | 1517.13 | 1556.78 | 1594.95 | 1628.52 | 1659.21 | 1684.53 | 1708.8 |

75 | 1499.75 | 1502.86 | 1530.74 | 1558.48 | 1597.72 | 1633.63 | 1664.01 | 1690.05 | 1717.91 | 1739.7 | 1758.34 |

80 | 1598.9 | 1605.38 | 1619.01 | 1646.45 | 1676.45 | 1707.29 | 1726.74 | 1749.67 | 1777.88 | 1794.69 | 1808.45 |

85 | 1703.25 | 1704.52 | 1709.66 | 1732.52 | 1757.97 | 1775.93 | 1794.13 | 1815.68 | 1833.76 | 1846.06 | 1858.25 |

90 | 1803.68 | 1797.91 | 1805.78 | 1825.43 | 1841.59 | 1849.65 | 1868.34 | 1881.64 | 1888.85 | 1899.29 | 1908.44 |

95 | 1905.7 | 1895.96 | 1907.78 | 1913.57 | 1918.23 | 1927.83 | 1934.13 | 1940.75 | 1945.3 | 1951.93 | 1956.47 |

100 | 2000 | 2000 | 2000 | 2000 | 2000 | 2000 | 2000 | 2000 | 2000 | 2000 | 2000 |

Agents Traversing Edge BC | |||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|

Travel Time Uncertainty ∆t | |||||||||||

P_{(NS)} | 0 | 10 | 20 | 30 | 40 | 50 | 60 | 70 | 80 | 90 | 100 |

0 | 4000 | 3949.61 | 3698.31 | 3469.51 | 3264.09 | 3095.23 | 2938.71 | 2803.68 | 2679.09 | 2590.05 | 2482.09 |

5 | 3800.14 | 3759.89 | 3517.36 | 3282.72 | 3095.49 | 2922.2 | 2768.52 | 2642.76 | 2514.28 | 2401.15 | 2325.51 |

10 | 3622.09 | 3548.39 | 3326.07 | 3110.78 | 2921.49 | 2758.76 | 2601.55 | 2484.67 | 2335.47 | 2241.89 | 2147.92 |

15 | 3416.16 | 3359.81 | 3161.58 | 2923.23 | 2753.44 | 2575.5 | 2430.58 | 2311.12 | 2203.9 | 2067.89 | 1972.54 |

20 | 3208.14 | 3166.32 | 2963.17 | 2763 | 2597.78 | 2398.26 | 2264.12 | 2160.07 | 2021.29 | 1918.45 | 1846.42 |

25 | 3011.63 | 2964.12 | 2772.76 | 2572.4 | 2393.1 | 2241.89 | 2096.7 | 1966.23 | 1866.79 | 1773.12 | 1677.57 |

30 | 2832.63 | 2769.42 | 2561.74 | 2398.05 | 2231.86 | 2093.81 | 1938.55 | 1845.89 | 1722.86 | 1618.63 | 1534.26 |

35 | 2588.06 | 2561.69 | 2385.22 | 2214.83 | 2085.4 | 1905.49 | 1789.16 | 1683.04 | 1589.84 | 1487.03 | 1415.04 |

40 | 2403.84 | 2353.68 | 2195.19 | 2051.78 | 1904.69 | 1759.28 | 1647.39 | 1550.42 | 1440.13 | 1350.45 | 1272.75 |

45 | 2179.75 | 2177.16 | 2019.88 | 1887.23 | 1744.3 | 1631.47 | 1496.15 | 1395.37 | 1304.91 | 1224.51 | 1138.18 |

50 | 1991.81 | 1988.52 | 1824.99 | 1703.99 | 1582.17 | 1462.94 | 1342.25 | 1258.84 | 1174.32 | 1096.34 | 1013.95 |

55 | 1796.01 | 1785.18 | 1655.9 | 1538.55 | 1447.95 | 1320.99 | 1217.91 | 1140.6 | 1052.52 | 970.6 | 919.93 |

60 | 1601.83 | 1596.3 | 1486.89 | 1404.63 | 1293.66 | 1175.4 | 1096.96 | 1007.02 | 930.6 | 865.28 | 803.88 |

65 | 1388.41 | 1389.98 | 1317.6 | 1231.07 | 1137.51 | 1039.24 | 957.87 | 873.18 | 808.16 | 751.88 | 696.36 |

70 | 1189.05 | 1193.43 | 1133.51 | 1050.97 | 965.78 | 886.51 | 810.15 | 743.02 | 681.62 | 631 | 582.46 |

75 | 1000.51 | 994.29 | 938.55 | 883.07 | 804.62 | 732.79 | 672.03 | 619.96 | 564.24 | 520.66 | 483.36 |

80 | 802.19 | 789.24 | 761.99 | 707.13 | 647.15 | 585.46 | 546.57 | 500.7 | 444.28 | 410.66 | 383.13 |

85 | 593.49 | 590.96 | 580.69 | 534.98 | 484.1 | 448.19 | 411.77 | 368.68 | 332.51 | 307.92 | 283.53 |

90 | 392.65 | 404.18 | 388.46 | 349.16 | 316.83 | 300.73 | 263.34 | 236.75 | 222.32 | 201.45 | 183.13 |

95 | 188.59 | 208.09 | 184.43 | 172.87 | 163.55 | 144.35 | 131.74 | 118.52 | 109.41 | 96.15 | 87.06 |

100 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |

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**Figure 1.**The simple network used to simulate the effect of cognitive agents and demonstrate Braess Paradox. By adding an edge (which should intuitively help) a negative impact on all users of a congested network can be observed. The latency function l of each edge with respect to the number of agents on the edge x is given accordingly.

**Figure 2.**Nash equilibrium flow and corresponding cost per agent C(f)/agent = 65 in the original network (

**left**), and Nash flow after insertion of a high capacity edge (

**right**). After adding the high capacity edge to the network, the total cost per agent C(f)/agent = 81, which is higher than in the original network.

**Figure 3.**Overview of the methodology to quantify the impact of cognitive agents, levels of selfishness and uncertainty on routing for a group of agents.

**Figure 4.**Diagram showing the latency values per agent for given non-selfishness probabilities over varying travel time uncertainty values ∆t.

**Figure 5.**Diagram showing the latency values per agent for given level of travel time uncertainty ∆t over varying probabilities of non-selfishness P

_{(NS)}.

**Table 1.**Average latency times $\overline{{C}^{\prime}}\left(f\right)$ of test runs for given values of varying ∆t and P

_{(NS)}.

Latency $\overline{{\mathit{C}}^{\prime}}{\left(\mathit{f}\right)}_{\mathsf{\Delta}\mathbf{t}{\mathit{P}}_{\left(\mathit{N}\mathit{S}\right)}}$ per Agent | |||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|

Travel Time Uncertainty ∆t | |||||||||||

P_{(NS)} | 0 | 10 | 20 | 30 | 40 | 50 | 60 | 70 | 80 | 90 | 100 |

0 | 81.00 | 80.55 | 78.40 | 76.58 | 75.05 | 73.88 | 72.86 | 72.02 | 71.29 | 70.80 | 70.22 |

5 | 79.25 | 78.91 | 76.95 | 75.19 | 73.88 | 72.75 | 71.81 | 71.09 | 70.39 | 69.81 | 69.43 |

10 | 77.78 | 77.19 | 75.50 | 73.99 | 72.75 | 71.75 | 70.86 | 70.23 | 69.48 | 69.04 | 68.62 |

15 | 76.17 | 75.75 | 74.33 | 72.76 | 71.72 | 70.72 | 69.95 | 69.37 | 68.87 | 68.28 | 67.89 |

20 | 74.66 | 74.37 | 73.01 | 71.78 | 70.84 | 69.79 | 69.14 | 68.67 | 68.09 | 67.68 | 67.42 |

25 | 73.33 | 73.02 | 71.84 | 70.70 | 69.77 | 69.04 | 68.40 | 67.87 | 67.49 | 67.16 | 66.84 |

30 | 72.20 | 71.82 | 70.64 | 69.79 | 68.99 | 68.39 | 67.76 | 67.41 | 66.99 | 66.66 | 66.41 |

35 | 70.78 | 70.64 | 69.73 | 68.92 | 68.35 | 67.63 | 67.21 | 66.86 | 66.57 | 66.28 | 66.09 |

40 | 69.82 | 69.57 | 68.83 | 68.21 | 67.63 | 67.11 | 66.74 | 66.45 | 66.15 | 65.93 | 65.75 |

45 | 68.76 | 68.75 | 68.08 | 67.56 | 67.06 | 66.70 | 66.30 | 66.04 | 65.82 | 65.65 | 65.48 |

50 | 67.97 | 67.95 | 67.34 | 66.93 | 66.55 | 66.21 | 65.91 | 65.72 | 65.55 | 65.41 | 65.27 |

55 | 67.24 | 67.20 | 66.77 | 66.42 | 66.17 | 65.86 | 65.64 | 65.49 | 65.33 | 65.21 | 65.14 |

60 | 66.61 | 66.59 | 66.28 | 66.06 | 65.80 | 65.55 | 65.41 | 65.26 | 65.15 | 65.07 | 65.00 |

65 | 66.02 | 66.03 | 65.85 | 65.66 | 65.48 | 65.31 | 65.19 | 65.08 | 65.01 | 64.95 | 64.91 |

70 | 65.58 | 65.59 | 65.47 | 65.33 | 65.20 | 65.10 | 65.01 | 64.95 | 64.90 | 64.87 | 64.84 |

75 | 65.25 | 65.24 | 65.16 | 65.09 | 65.00 | 64.94 | 64.89 | 64.86 | 64.83 | 64.82 | 64.81 |

80 | 65.00 | 64.99 | 64.96 | 64.92 | 64.88 | 64.84 | 64.83 | 64.81 | 64.80 | 64.80 | 64.80 |

85 | 64.85 | 64.85 | 64.84 | 64.82 | 64.81 | 64.80 | 64.80 | 64.80 | 64.81 | 64.81 | 64.82 |

90 | 64.80 | 64.80 | 64.80 | 64.80 | 64.81 | 64.81 | 64.82 | 64.83 | 64.84 | 64.85 | 64.86 |

95 | 64.86 | 64.85 | 64.86 | 64.86 | 64.87 | 64.88 | 64.89 | 64.90 | 64.91 | 64.92 | 64.92 |

100 | 65.00 | 65.00 | 65.00 | 65.00 | 65.00 | 65.00 | 65.00 | 65.00 | 65.00 | 65.00 | 65.00 |

**Table 2.**Lowest average latency times $\overline{{C}^{\prime}}\left(f\right)$ for varying ∆t and P

_{(NS)}are marked with orange colored numbers.

Latency | $\overline{{\mathit{C}}^{\prime}}{\left(\mathit{f}\right)}_{\mathsf{\Delta}\mathbf{t}{\mathit{P}}_{\left(\mathit{N}\mathit{S}\right)}}$ per Agent | ||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|

Travel Time Uncertainty ∆t | |||||||||||

P_{(NS)} | 0 | 10 | 20 | 30 | 40 | 50 | 60 | 70 | 80 | 90 | 100 |

0 | 81.00 | 80.55 | 78.40 | 76.58 | 75.05 | 73.88 | 72.86 | 72.02 | 71.29 | 70.80 | 70.22 |

… | … | … | … | … | … | … | … | … | … | … | … |

… | … | … | … | … | … | … | … | … | … | … | … |

80 | 65.00 | 64.99 | 64.96 | 64.92 | 64.88 | 64.84 | 64.83 | 64.81 | 64.80 | 64.80 | 64.80 |

85 | 64.85 | 64.85 | 64.84 | 64.82 | 64.81 | 64.80 | 64.80 | 64.80 | 64.81 | 64.81 | 64.82 |

90 | 64.80 | 64.80 | 64.80 | 64.80 | 64.81 | 64.81 | 64.82 | 64.83 | 64.84 | 64.85 | 64.86 |

95 | 64.86 | 64.85 | 64.86 | 64.86 | 64.87 | 64.88 | 64.89 | 64.90 | 64.91 | 64.92 | 64.92 |

100 | 65.00 | 65.00 | 65.00 | 65.00 | 65.00 | 65.00 | 65.00 | 65.00 | 65.00 | 65.00 | 65.00 |

**Table 3.**Correlation % between $\overline{{C}^{\prime}}\left(f\right)$ for given ∆t levels and P

_{(NS)}, which indicates the dependence of travel delay on the uncertainty in travel information.

$\overline{{\mathit{C}}^{\prime}}\left(\mathit{f}\right)\mathbf{on}\Delta \mathit{t}$ | |
---|---|

${\mathsf{\rho}}_{{\Delta}_{\mathrm{t}}}$ | |

P_{(NS)} 1 | 1 |

∆t = 0 | −0.947 |

∆t = 10 | −0.947 |

∆t = 20 | −0.944 |

∆t = 30 | −0.941 |

∆t = 40 | −0.936 |

∆t = 50 | −0.926 |

∆t = 60 | −0.918 |

∆t = 70 | −0.911 |

∆t = 80 | −0.902 |

∆t = 90 | −0.890 |

∆t = 100 | −0.880 |

**Table 4.**Correlation % between for given P

_{(NS)}levels and ∆t, which indicates the dependence between travel delay and the probability of selfishness.

$\overline{{C}^{\prime}}\left(f\right)$ on P_{(NS)} | |
---|---|

${\mathbf{\rho}}_{{\mathrm{P}}_{\left(NS\right)}}$ | |

∆t | 1 |

P_{(NS)} = 0 | −0.980 |

P_{(NS)} = 5 | −0.980 |

P_{(NS)} = 10 | −0.982 |

P_{(NS)} = 15 | −0.982 |

P_{(NS)} = 20 | −0.981 |

P_{(NS)} = 25 | −0.980 |

P_{(NS)} = 30 | −0.981 |

P_{(NS)} = 35 | −0.981 |

P_{(NS)} = 40 | −0.982 |

P_{(NS)} = 45 | −0.982 |

P_{(NS)} = 50 | −0.979 |

P_{(NS)} = 55 | −0.982 |

P_{(NS)} = 60 | −0.982 |

P_{(NS)} = 65 | −0.982 |

P_{(NS)} = 70 | −0.978 |

P_{(NS)} = 75 | −0.972 |

P_{(NS)} = 80 | −0.956 |

P_{(NS)} = 85 | −0.746 |

P_{(NS)} = 90 | 0.963 |

P_{(NS)} = 95 | 0.984 |

P_{(NS)} = 100 | N/A |

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## Share and Cite

**MDPI and ACS Style**

Scholz, J.; Church, R.L.
Shortest Paths from a Group Perspective—A Note on Selfish Routing Games with Cognitive Agents. *ISPRS Int. J. Geo-Inf.* **2018**, *7*, 345.
https://doi.org/10.3390/ijgi7090345

**AMA Style**

Scholz J, Church RL.
Shortest Paths from a Group Perspective—A Note on Selfish Routing Games with Cognitive Agents. *ISPRS International Journal of Geo-Information*. 2018; 7(9):345.
https://doi.org/10.3390/ijgi7090345

**Chicago/Turabian Style**

Scholz, Johannes, and Richard L. Church.
2018. "Shortest Paths from a Group Perspective—A Note on Selfish Routing Games with Cognitive Agents" *ISPRS International Journal of Geo-Information* 7, no. 9: 345.
https://doi.org/10.3390/ijgi7090345