Complex Systems in Industrial Mathematics Dr Robert Leese Smith Institute Bath Institute for Complex Systems 31 January 2005

2 Consumer behaviour  Purchasing decisions are a function not only of individual products, but also of the consumer, in particular their reaction to competing products - psychology, and their social environment of other consumers - sociology. Strong networking, no advertising Weaker networking, limited advertising

3 Task allocation  A collection of players each has a task to allocate to one of m machines. Each machine services the tasks that are assigned to it in random order.  Consider three players, with player i having a task of length i ( i =1,2,3), and two machines.  The optimal solution in terms of minimum makespan is for players 1 and 2 to select machine 1 and player 3 to select machine 2.  However, there are other Nash equilibria (corresponding to sets of strategies in which no player has an incentive to change). Machine 1 Machine 2 Optimal allocation Prob 1/2

4 The Price of Anarchy  The players may adopt mixed strategies, in which tasks are assigned at random to machines. One equilibrium is for the players to make allocations to the two machines with the following probabilities: u Player 1: (1,0) u Player 2: (1/4, 3/4) u Player 3: (1/3, 2/3)  The expected makespan is now 9/2, which is a factor of 3/2 greater than the best possible value.  The Price of Anarchy is the worst-case (i.e. maximum) ratio of the makespan of a Nash equilibrium and the optimal makespan, taken over all possible collections of tasks. For allocations to two machines, it is indeed 3/2. For m machines, it is  (log m / log log m ).

5  E.g. Braess Paradox (again with unit total flow)  The price of anarchy in both of these examples is 4/3, which is the maximum possible for networks with linear latency functions. Flow networks  Consider a network with a source A and sink B, with each link incurring a latency that in general depends on the flow through it.  E.g. Parallel links with unit total flow: AB l(f)=1 l(f)=f A C D B l(f)=1 l(f)=f l(f)=0

6 A mechanical analogue of the Braess paradox  Think of the support as the source and the weight as the sink.  Strings corresponds to links with constant latency and springs to links with linear latency.  No collection of severed strings and springs can raise the weight by more than D/4 (Roughgarden and Tardos, 2002). D

7 Designing systems with good properties  Task allocation problems: u Different scheduling policies (coordination mechanisms) among machines can reduce the price of anarchy, e.g. one machine processing tasks in order of increasing length and one in order of decreasing length (the price of anarchy then becomes 1 for three players).  Flow problems: u Introducing taxes (or tolls) on each link produces a cost that consists of latency plus tax. The price of anarchy is now the worst-case ratio of total cost and optimal latency, taken over all source-sink flow rates. u For the Braess network, a toll of 1 on link DC whenever the flow on that link is at least 2/3 (and zero otherwise) leads to a price of anarchy equal to 16/15.

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