A theory of game trees, based on solution trees

In this paper a complete theory of game tree algorithms is presented, entirely based upon the notion of a solution tree. Two types of solution trees are distinguished: max and min solution trees respectively. We show that most game tree algorithms construct a superposition of a max and a min solution tree. Moreover, we formulate a general cut-off criterion in terms of solution trees. In the second half of this paper four well known algorithms, viz., alphabeta, SSS*, MTD and Scout are studied extensively. We show how solution trees feature in these algorithms and how the cut-off criterion is applied

From game trees to a narrative game: prototype

The game theory started as a mathematical exploration of human behaviours. The term has expanded to cover this mathematical exploration of the decision-making not only of humans but also of animals and computers. Through this project, we will be able to study how can we integrate the mathematical aspect of a specific topic from game theory into a development of a game. Furthermore, what advantages or inconveniences can we find on our way to integrating this field into game development? First, we must focus on learning key concepts about game theory topics, specifically Game Trees and subgame perfect equilibrium. These concepts will help us create a game tree and calculate the decision-making values, so later, it becomes easier to mix with other concepts once we accomplish the research. When we look at a game tree, we can easily relate some similarities with another well-known video game system. These systems are narrative trees or decision trees. So the main focus of this project is to create a game prototype where a player has to make decisions like in a narrative game. However, instead of a standard decision tree, we will investigate making this tree with the concepts learned through game theory. The key to success will be learning how to create a game tree and combining this with some psychology so players can feel that their decisions have some value, as we would see in a game tree. All these challenges will illustrate through the development part of the project

Game tree algorithms and solution trees

In this paper, a theory of game tree algorithms is presented, entirely based upon the concept of solution tree. Two types of solution trees are distinguished: max and min trees. Every game tree algorithm tries to prune nodes as many as possible from the game tree. A cut-off criterion in terms of solution trees will be formulated, which can be used to eliminate nodes from the search without affecting the result. Further, we show that any algorithm actually constructs a superposition of a max and a min solution tree. Finally, we will see, how solution trees and the related cutoff criterion are applied in major game tree algorithms, like alpha-beta and MTD

Chain models, trees of singular cardinality and dynamic EF games

Let κ be a singular cardinal. Karp's notion of a chain model of size ? is defined to be an ordinary model of size κ along with a decomposition of it into an increasing union of length cf(κ). With a notion of satisfaction and (chain)-isomorphism such models give an infinitary logic largely mimicking first order logic. In this paper we associate to this logic a notion of a dynamic EF-game which gauges when two chain models are chain-isomorphic. To this game is associated a tree which is a tree of size κ with no κ-branches (even no cf(κ)-branches). The measure of how non-isomorphic the models are is reflected by a certain order on these trees, called reduction. We study the collection of trees of size κ with no κ-branches under this notion and prove that when cf(κ) = ω this collection is rather regular; in particular it has universality number exactly κ+. Such trees are then used to develop a descriptive set theory of the space cf(κ)κ.The main result of the paper gives in the case of κ strong limit singular an exact connection between the descriptive set-theoretic complexity of the chain isomorphism orbit of a model, the reduction order on the trees and winning strategies in the corresponding dynamic EF games. In particular we obtain a neat analog of the notion of Scott watershed from the Scott analysis of countable models

Strategic Network Formation with Attack and Immunization

Strategic network formation arises where agents receive benefit from connections to other agents, but also incur costs for forming links. We consider a new network formation game that incorporates an adversarial attack, as well as immunization against attack. An agent's benefit is the expected size of her connected component post-attack, and agents may also choose to immunize themselves from attack at some additional cost. Our framework is a stylized model of settings where reachability rather than centrality is the primary concern and vertices vulnerable to attacks may reduce risk via costly measures. In the reachability benefit model without attack or immunization, the set of equilibria is the empty graph and any tree. The introduction of attack and immunization changes the game dramatically; new equilibrium topologies emerge, some more sparse and some more dense than trees. We show that, under a mild assumption on the adversary, every equilibrium network with $n$ agents contains at most $2n-4$ edges for $n\geq 4$. So despite permitting topologies denser than trees, the amount of overbuilding is limited. We also show that attack and immunization don't significantly erode social welfare: every non-trivial equilibrium with respect to several adversaries has welfare at least as that of any equilibrium in the attack-free model. We complement our theory with simulations demonstrating fast convergence of a new bounded rationality dynamic which generalizes linkstable best response but is considerably more powerful in our game. The simulations further elucidate the wide variety of asymmetric equilibria and demonstrate topological consequences of the dynamics e.g. heavy-tailed degree distributions. Finally, we report on a behavioral experiment on our game with over 100 participants, where despite the complexity of the game, the resulting network was surprisingly close to equilibrium.Comment: The short version of this paper appears in the proceedings of WINE-1