*Friday, August 2, 10:10 a.m. - 12:00 p.m., Duke Energy Convention Center, Room 200*

We encounter uncertainty everywhere, at all levels of our consciousness, in almost every one of our endeavors. Even things of which we are certain: the sun will rise tomorrow, our current existence has a finite time span, are subject to imprecision. How has mathematics helped us understand uncertainty and unpredictability? How can we use use quantitative tools to make decisions under incomplete information or cognitive limitations?

In this session we will present mathematical tools and results from probability, dynamical systems and ergodic theory that give insight into these questions.

**Organizer:**

**Ami Radunskaya**, *Pomona College*

### Crossing the Threshold: The Role of Demographic Stochasticity in the Evolution of Cooperation

*10:10 a.m. - 10:30 a.m.*

**Tom LoFaro**, *Gustavus Adolphus College*

#### Abstract

We will discuss a pair of models for the evolution of cooperation that incorporates game theoretic ideas into a discrete generation population genetics framework. The first model will be completely deterministic and we will show that when a small population of "cooperators" is included into a larger population of "defectors" then the cooperators will always die off. However, when stochasticity is incorporated into the model then there is the potential for the population of cooperators to become established. We will explore conditions that increase the likelihood that this will occur. In particular, we will prove that if the relative advantage of defecting is small relative to the benefits of cooperating then it is more probable that cooperation will evolve.

### Stochastic Perturbations of the Logistic Map

*10:40 a.m. - 11:00 a.m.*

**Kim Ayers**, *Pomona College*

#### Abstract

The logistic map is a famed map on [0,1] that exhibits a period-doubling bifurcation that eventually leads to a chaotic regime. In this talk, we examine a stochastically perturbed logistic map, and the dynamics that arise. We’ll examine the differences in dynamics when looking at uniformly distributed noise, as compared with noise that is distributed according to a β-distribution. In this situation, fixed points and period orbits are no longer deterministic objects, but rather are random variables. How does this affect the chaotic regime?

### Logic for Reasoning about Uncertainty Dynamics and Informational Cascades

*11:10 a.m. - 11:30 a.m.*

**Joshua Sack**, *California State University, Long Beach*

#### Abstract

Uncertainty of a group of people or agents can be represented using a multi-graph, where the vertices are the possibilities and agent-labeled edges reflect uncertainty agents have between the possibilities. Informational events change these graphs to reflect new states of uncertainty. This talk discusses how such dynamics is modeled and described by probabilistic dynamic epistemic logics, and discusses how such logics can help us reason about informational cascade, a phenomenon where agents prioritize judgements by other observers over their direct observations.

### Probability As a Tool for Studying Problems in Behavioral Economics

*11:40 a.m. - 12:00 p.m.*

**Aloysius Bathi Kasturiarachi**, *Kent State University*

#### Abstract

In this presentation we introduce probabilistic ideas to study problems in behavioral economics. The research will have three components that form the foundational props of the platform on which probability will uncover the rich interplay between mathematics and behavioral economics.

The first component explores through several probability experiments, how Deterministic Behavior can be extended to understand Uncertainty and Misbehavior. We evaluate the number of games one will need to play (against the dealer) in order to win a prize, if a priori, the player knows that the probability of winning is less than 0.5. Even though at the outset the player is destined to lose, a calculation can be done to come up with the ideal number of plays for the game, to maximize chances of winning if the player’s probability is close to but less than 0.5. This idea can be further extended when a financial incentive is added to each extra game the player is willing to risk. The second component is to turn Uncertainty into Relative Determinism using probabilistic ideas. The examples for this study comes from financial markets. The final component involves experiments performed with human subjects that provides insights to refining the *value function model* in behavioral economics. This is an example on how we learn by *learning more* in high-stakes complex systems, such as medicine.