Describe how the minimax and alpha–beta algorithms change for two-player, non-zero-sum games in which each player has a distinct utility function and both utility functions are known to both players. If there are no constraints on the two terminal utilities, is it possible for any node to be pruned by alpha–beta? What if the player’s utility functions on any state differ by at most a constant k , making the game almost cooperative?

We can implement requests to the waiter as either a queue of requests or as a periodic retry of a request. With a queue, requests are handled in the order they are received. Th e problem with using the queue is that we may not always be able to service the philosopher whose request is at the head of the queue (due to the unavailability of resources). Describe a scenario with 5 philosophers where a queue is provided, but service is not granted even though there are forks available for another philosopher (whose request is deeper in the queue) to eat.If we implement requests to the waiter by periodically repeating our request until the resources become available, will this solve the problem described in the above Exercise? Explain.

Exercise 3 Consider calculation of PageRank for a directed graph. Consider two jumping probabilities a and B such that a > B. For every node u, let Pa(u) be the PageRank of u calculated using a and let Pg (u) be the PageRank of u calculated using B. For each of the following statements, indicate whether the statement is true or false and provide a brief explanation for your answer. 1. The inequality Pa (u) > Pg (u) must hold for all nodes u. 2. The inequality Pa(u) > Pg (u) must hold for at least one node u.

3. Suppose that we want to know which agents are connected by a walk of length two in a given network. That is, want to know the set of ordered pairs (i, j) such that there exists a walk of length 2 between any i and j. Define n as the number of nodes and m as the number of links. (a) One way to do this is to input the adjacency matrix A, calculate A², then output all pairs (i, j) for which A(i) > 0. Defining scalar multiplication and addition as "basic operations," up to a constant c, how many "basic operations" are required to calculate A², in terms of n and m? (Hint: First think about how many operations are required to calculate each entry of A², then multiply by the number of entries.) (b) Suppose that we want to know the number of agents who are connected by a walk of length k in a given network, where k≥ 1. Up to a constant, how many basic operations are required to calculate Ak, in terms of n and m? (c) (Harder) We showed in class (Lecture 4) that a simple adaptation of…