# Agreement Technologies for Coordination in Smart Cities
## 3 Applications
### Coordination of Traffic Flows through Intelligent Intersections
#### 3.1.2 Mechanisms for Multiple Intersections
Then, each intersection manager adjusts the prices \(p^{t}(l)\) for all the incoming links, lowering them if there is excess supply (\(z^{t}(l)<0\)) and raising them if there is excess demand (\(z^{t}(l)>0\)). The new price vector \(p^{t+1}\) is communicated to the drivers that iteratively choose their new desired route on this basis. Once the equilibrium price is computed, the trading transactions take place and each driver buys the required reservations at the intersections that lay on its route. In order to adapt the Walrasian auction to the traffic domain, we implemented a pricing strategy that aims at reaching the equilibrium price but works on a continuous basis, with drivers that leave and join the market dynamically, and with transactions that take place continuously. To reach general equilibrium, each intersection manager applies the following _price update rule_: at time \(t\), it independently computes the excess demand \(z^{t}(l)\) and updates the price \(p^{t}(l)\) as follows:  
\[p^{t+1}(l)=max\left[\delta,p^{t}(l)+p^{t}(l)\frac{z^{t}(l)}{s(l)}\right]\]  
where \(\delta\) is the minimum price that the intersection manager charges for the reservations that it sells. As drivers that travel through road network links with low demand shall not incur any costs, for the CTA strategy, we choose \(\delta=0\). The _integrated mechanism for traffic management_ (CA-CTA) combines the competitive traffic assignment strategy (CTA) with the auction-based policy (CA). Since the intersection manager is the supplier of the reservations that are allocated through the combinatorial auction, it may control the _reserve price_ of the auctioned reservations, that is, the minimum price at which the intersection manager is willing to sell. At time \(t\), for each link \(l\), CTA-CA simply sets this reserve price to the price \(p^{t}(l)\) computed by the price update rule of the CTA strategy. The experimental evaluation of the strategies was performed on a hybrid mesoscopic-microscopic simulator, where traffic flow on road segments was modelled at mesoscopic level, while traffic flow inside intersections was modelled at microscopic level. Although our work does not depend on the underlying road network, we chose a topology inspired by the urban road network of the city of Madrid for our empirical evaluation (see Figure 6). The network is characterised by several freeways that connect the city centre with the surroundings and a ring road. Each large dark vertex in Figure 6, if it connects three or more links, is modelled as a reservation-based intersection. In the experiments, we recreated a typical high-load situation (i.e., the central, worst part of a morning peak), with more than 11,000 vehicles departing within a time window of 50 min (see [27] for details). We aimed at comparing the performance of FCFS, CTA, and CTA-CA. In FCFS, each intersection manager performs combinatorial auctions (without reserve price) in isolation. In this case, the drivers' route choice model simply selects the route with minimum expected travel time at free flow, since there is no notion of price. For the other strategies, we assumed that drivers choose the most preferred route they can afford. Since the prices of links are changing dynamically, a driver continuously evaluates the utility of the route it is following and, in the case that a different route becomes more attractive, it may react and change on-the-fly how to reach its destination, selecting a route different from the original one. To assess the social cost incurred by CA-CTA at the global level, we measured the moving average of the travel time, that is, how the average travel time of the entire population of drivers, computed over all the O-D pairs, evolves during the simulation. The results, with 95% confidence interval error bars, are plotted in Figure 7. In the beginning, the average travel time is similar for all the scenarios, but as the number of drivers that populate the network (i.e., its load) increases, it grows significantly faster with FCFS than with the CA-CTA policy.