A search game on a cyclic graph

Abstract

There is a finite cyclic graph. The hider chooses one of all nodes except the specified one, and he hides an (immobile) object there. At the beginning the seeker is at the specified node. After the seeker chooses an ordering of the nodes except the specified one, he examines each nodes in that order until he finds the object, traveling along edges. It costs an amount when he moves from a node to an adjacent one and also when he checks a node. While the hider wishes to maximize the sum of the traveling costs and the examination costs which are required to find the object, the seeker wishes to minimize it. The problem is modeled as a two-person zero-sum game. We solve the game when unit costs (traveling cost ϩ examination cost) have geometrical relations depending on nodes. Then we give properties of optimal strategies of both players.

Search Games: Literature and Survey

“…This game is a combination of the rendezvous search and the search game. In addition to the games mentioned so far, Cao (1995) [55], Alpern (2008Alpern ( , 2010 [6,7] and Dagan and Gal (2008) [63] studied HSGs defined on trees, and Pavlovic (1995) [190], Kikuta (2004) [151], [13] and Alpern and Baston (2009) [12] discussed HSGs on networks.…”

Search games on networks with travelling and search costs and with arbitrary searcher starting points

“…Because the costs incurred by the searcher are made up of the distance travelled and the search costs at nodes, the searcher strategy space can be represented by a permutation of the nodes which represents the order in which they should be searched with the understanding that the searcher will move from one node to the next using a path of minimum length between them. This is the searcher strategy space used in Kikuta [8] and Kikuta [9] for games in which the searcher is forced to start at a designated node. However, for the arguments in the next section, it is more convenient to have the path taken by the searcher occurring explicitly in the definition of searcher strategy.…”

“…N = {1,2,…, n } and edge set E = ( n ,1) ∪∪ ( i , i + 1); in an optimal strategy, the searcher selects (with appropriate probabilities) a starting node and a direction of travel and simply searches at each node he visits. This simple solution is in stark contrast to the game on the cycle network where the searcher has a designated starting point (see Kikuta [9]). Note that other special cases of the theorem are the complete network and the complete bipartite network ( m , m ).…”

“…When the searcher reaches a node, he does not have to search it but can pass through it and return at a later time to conduct a search if he has not found the hider in the meantime. In these games, the searcher not only has travelling costs but also search costs; for papers on this type of problem when the searcher has a designated starting point see Kikuta and Ruckle [10], Kikuta [8] and Kikuta [9].…”

Search Games on a Broken Wheel With Traveling and Search Costs

“…Even more recently Hohzaki [12] has produced a very comprehensive survey of search games. When the hider is constrained to hide at a node it is usual to include an inspection or search cost so that the searcher has both travelling and search costs; problems of this type in which the searcher starts from a designated node are analysed in Kikuta and Ruckle [15], Kikuta [13] and Kikuta [14]. More recently there have been papers in which the searcher can choose the starting node (see Alpern, Baston and Gal [1] and [2], Dagan and Gal [9], Baston and Kikuta [7] and Baston and Kikuta [8]); in the first three of these papers the hider can hide at any point of the network whereas, in the last two, the hider must hide at a node.…”