There are a number of domains where agents must collectively form a network in the face of the following trade-off: each agent receives benefits from the direct links it forms to others, but these links expose it to the risk of being hit by a cascading failure that might spread over multistep paths. Financial contagion, epidemic disease, and the exposure of covert organizations to discovery are all settings in which such issues have been articulated.Here we formulate the problem in terms of strategic network formation, and provide asymptotically tight bounds on the welfare of both optimal and stable networks. We find that socially optimal networks are, in a precise sense, situated just beyond a phase transition in the behavior of the cascading failures, and that stable graphs lie slightly further beyond this phase transition, at a point where most of the available welfare has been lost. Our analysis enables us to explore such issues as the trade-offs between clustered and anonymous market structures, and it exposes a fundamental sense in which very small amounts of "over-linking" in networks with contagious risk can have strong consequences for the welfare of the participants.

Gamification is growing increasingly prevalent as a means to incentivize user engagement of social media sites that rely on user contributions. Badges, or equivalent rewards, such as top-contributor lists that are used to recognize a user's contributions on a site, clearly appear to be valued by users who actively pursue and compete for them. However, different sites use different badge designs, varying how, and for what, badges are awarded. Some sites, such as StackOverflow, award badges for meeting fixed levels of contribution. Other sites, such as Amazon and Y! Answers, reward users for being among some top set of contributors on the site, corresponding to a competitive standard of performance. Given that users value badges, and that contributing to a site requires effort, how badges are designed will affect the incentives-therefore the participation and effort-elicited from strategic users on a site.We take a game-theoretic approach to badge design, analyzing the incentives created by widely used badge designs in a model in which winning a badge is valued, effort is costly, and potential contributors to the site endogenously decide whether or not to participate, and how much total effort to put into their contributions to the site. We analyze equilibrium existence, as well as equilibrium participation and effort, in an absolute standards mechanism M α in which badges are awarded for meeting some absolute level of (observed) effort, and a relative standards mechanism M ρ corresponding to competitive standards, as in a top-ρ contributor badge. We find that equilibria always exist in both mechanisms, even when the value from winning a badge depends endogenously on the number of other winners. However, M α has zero-participation equilibria for standards that are too high, whereas all equilibria in M ρ elicit nonzero participation for all possible ρ, provided that ρ is specified as a fixed number rather than as a fraction of actual contributors (note that the two are not equivalent in a setting with endogenous participation). Finally, we ask whether or not a site should explicitly announce the number of users winning a badge. The answer to this question is determined by the curvature of the value of winning the badge as a function of the number of other winners.