We propose a definition of mass for characteristic hypersurfaces in
asymptotically vacuum space-times with non-vanishing cosmological constant
Λ ∈ R
, generalising the definition of Trautman and Bondi for Λ = 0.
We show that our definition reduces to some standard definitions in several
situations. We establish a balance formula linking the characteristic
mass and a suitably defined renormalised volume of the null hypersurface,
generalising the positivity identity of one of us (PTC) and Paetz proved
when Λ = 0.
We study evolutionary games on graphs. The individuals of a population
occupy the vertices of the graph and interact with their neighbors to receive
payoff. We consider finite population size, regular graphs, probabilistic
death-birth updating and weak selection. There are two types of strategies,
A and B, and a payoff matrix [(a, b),(c, d)]. The initial condition is given by
an arbitrary configuration where each vertex is occupied by either A or B.
The conjugate initial condition is obtained by swapping A and B. We ask:
when is the fixation probability of A for the original configuration greater
than the fixation probability of B for the conjugate configuration? The answer
is a linear condition of the form σa+b > c+σd. We calculate σ for any
initial condition. For large population size we obtain the well known result
σ = (k + 1)/(k − 1), but now this result extends to any mixed initial condition.
As a specific example we study evolution of cooperation. We calculate
the critical benefit-to-cost ratio for natural selection to favor the fixation of
cooperators for any initial condition. We obtain results that specify which
initial conditions reduce and which initial conditions increase the critical
benefit-to-cost ratio. Adding more cooperators to the initial condition does
not necessarily favor cooperation. But strategic placing of cooperators in a
network can enhance the takeover of cooperation.
We provide a model of a decentralized, dynamic auction market platform (e.g.,
eBay) in which a large number of buyers and sellers participate in simultaneous, singleunit
auctions each period. Our model accounts for the endogenous entry of agents and
the impact of intertemporal optimization on bids. Solving our model with a finite number
of bidders is computationally intractable due to the curse of dimensionality, so we prove
that a continuum version of our model provides a good approximation of an equilibrium
in the finite model. We use the approximation to estimate the structural primitives of our
model using Kindle sales on eBay. We find that just over one third of Kindle auctions on
eBay result in an inefficient allocation with deadweight loss amounting to 13.5% of total
possible market surplus. We also find that partial centralization - for example, runnng
half as many 2-unit, uniform price auctions each day - would eliminate a large fraction
of the inefficiency, but yield lower seller revenues. Our results highlight the importance
of understanding platform composition effects—selection of agents into the market—in
assessing the implications of market design.
We present a simple model for the evolution of social behavior in family-structured, finite sized
populations. Interactions are represented as evolutionary games describing frequency-dependent
selection. Individuals interact more frequently with siblings than with members of the general
population, as quantified by an assortment parameter r, which can be interpreted as “relatedness”.
Other models, mostly of spatially structured populations, have shown that assortment can promote the
evolution of cooperation by facilitating interaction between cooperators, but this effect depends on the
details of the evolutionary process. For our model, we find that sibling assortment promotes cooperation
in stringent social dilemmas such as the Prisoner's Dilemma, but not necessarily in other situations.
These results are obtained through straightforward calculations of changes in gene frequency. We also
analyze our model using inclusive fitness. We find that the quantity of inclusive fitness does not exist for
general games. For special games, where inclusive fitness exists, it provides less information than the