Stackelberg pricing games with congestion effects

We study a Stackelberg game with multiple leaders and a continuum of followers that are coupled via congestion effects. The followers’ problem constitutes a nonatomic congestion game, where a population of infinitesimal players is given and each player chooses a resource. Each resource has a linear cost function which depends on the congestion of this resource. The leaders of the Stackelberg game each control a resource and determine a price per unit as well as a service capacity for the resource influencing the slope of the linear congestion cost function. As our main result, we establish existence of pure-strategy Nash–Stackelberg equilibria for this multi-leader Stackelberg game. The existence result requires a completely new proof approach compared to previous approaches, since the leaders’ objective functions are discontinuous in our game. As a consequence, best responses of leaders do not always exist, and thus standard fixed-point arguments á la Kakutani (Duke Math J 8(3):457–458, 1941) are not directly applicable. We show that the game is C-secure (a concept introduced by Reny (Econometrica 67(5):1029–1056, 1999) and refined by McLennan et al. (Econometrica 79(5):1643–1664, 2011), which leads to the existence of an equilibrium. We furthermore show that the equilibrium is essentially unique, and analyze its efficiency compared to a social optimum. We prove that the worst-case quality is unbounded. For identical leaders, we derive a closed-form expression for the efficiency of the equilibrium.

Is Voting for a Cartel a Sign of Cooperativeness?

This paper tests the hypothesis that a (partial) reason why cartels—collective but costly and non-binding price agreements—lead to higher prices in a Bertrand oligopoly could be because of a selection effect: decision-makers who are willing to form price agreements are more likely to be less competitive and pick higher prices in general. To test this hypothesis we run an experiment where participants play two consecutive Bertrand pricing games: first a standard version without the opportunity to form agreements; followed by a version where participants can vote whether to have a (costly) non-binding agreement as a group to pick the highest number. We find no statistically significant difference between the numbers picked in the first game by participants who vote for and against an agreement in the second game. We do confirm that having a non-binding agreement to cooperate leads to higher numbers being picked on average. Both participants who voted for and against the agreement increase the number they pick in situations with an agreement. However, this effect is bigger for participants who voted in favour.

Network Pricing: How to Induce Optimal Flows Under Strategic Link Operators

Regulating Network Pricing Games by Using Price Caps

On pricing and bundling decisions for Stackelberg games in parallel channels of substitutable composites

The paper describes competition within a supply network with parallel distribution channels. Each supply chain in the network is composed of a manufacturer and a retailer. Manufacturers sell two complementary products to the retailers, who then deliver to the end consumers. All players can bundle or not bundle their products assuming that the retail market presents the products in a mixed bundling setting. The motivation of this study is to mainly analyze the impact of cost reduction via manufacturers, on how the whole supply network will behave. We have modeled and solved partly and fully sequential game structures well known as Bertrand and Stackelberg games, where the preceding movers are considered to have more market power. Mathematical and numerical analyses reveal interesting propositions and managerial insights for decision makers who are practicing cost cutting strategies. The combination of different ordinal structures have led to exact mathematical comparisons among 24 games. Results indicate both manufacturers and retailers are better off with simultaneous pricing games which promotes the concept of coordination through layer and channels of the network. Cost reduction with compensation increases payoffs, if only applied by the manufacturer whose complementary products’ manufacturing costs are more distanced. It is also shown that retailers enjoy a retail advantage on one product at its best when playing retailer leading Stackelberg games.