Optimal cache search depends on precision of spatial memory and pilfering, but what if that knowledge is not perfect? G. Pfuhl a , H. Tjelmeland b,1 , S. Molden c, 2 , R. Biegler a, * a Psykologisk institutt, Norges teknisk-naturvitenskapelige universitet (NTNU) b Institutt for matematiske fag, Norges teknisk-naturvitenskapelige universitet (NTNU) c Centre for the Biology of Memory, Medical–Technical Research Centre, Norges teknisk-naturvitenskapelige universitet (NTNU) article info Article history: Received 24 September 2008 Initial acceptance 29 December 2008 Final acceptance 11 June 2009 Published online 11 August 2009 MS. number: 08-00617R Keywords: caching cognition precision spatial memory uncertainty The problem of when an action should be abandoned because it is not worth further effort occurs in many situations. In the spatial domain, the relevant information can be quantified. Two essential pieces of information are the precision with which the target location and the probability of the target being present are known. We offer a quantitative description of optimal solutions to cache retrieval, treating it as a 2D investment problem with search cost proportional to area. We estimated the value of knowing the precision of spatial information and the precision of information regarding probability: how precisely should you estimate the precision of your knowledge? We compared the expected gain from assessing the precision of spatial knowledge and probability with the expected gain from decisions based on aggregate knowledge of the distribution of precision and probability. We found that heuristics, repre- sented here as default search limits based on aggregate knowledge, are useful only under limited conditions. Ó 2009 The Association for the Study of Animal Behaviour. Published by Elsevier Ltd. All rights reserved. You are courting, but the object of your desire does not respond. Does this mean you should try harder, or that you should give up? You have started a new business, which is still losing money. Do you persevere, or try something else? You direct the aerial search for the crew of a sunken ship, at the position reported in the Mayday call. At what point do you decide that failure to find the crew means not that the position was wrong, but that the crew must have gone down with the ship? A squirrel searches for a cache of buried nuts. At what point should it stop digging, effectively assuming the cache has been pilfered, and go elsewhere? The question of how much to invest in an activity or project and when to abandon it is fairly general, but often the optimal course of action is difficult to determine. One may be content with merely making a good choice, but to analyse the trade-offs between the benefits of a better choice and the costs of acquiring the necessary information, one needs to know the optimum and the conse- quences of deviating from the optimum. The spatial version of the problem, when to abandon search, is a mathematically tractable case of an investment problem. The relevant information, namely the precision of spatial information and the probability of the target being present at all, can be quantified. The effect this information should have on decisions is equally quantifiable. We use the squirrel’s cache retrieval as our example problem. We analyse that in moderate depth before returning to the more general case of investment problems, and discuss differences to other models of foraging and search. THE CACHE RETRIEVAL PROBLEM The initial precision with which the squirrel stores the location of its cache in memory can be characterized by a probability distribu- tion, and forgetting by the broadening of that distribution (White & Wixted 1999; for empirical data see Perkins & Weyant 1958; Thomas & Lopez 1962; Cheng et al. 1997; Biegler et al. 2001). More specifi- cally, we assume that knowledge of the location of a hidden item can be characterized by a normal probability density distribution. Uncertainty regarding location is represented by the standard deviation of the normal distribution. Precision is the inverse of the standard deviation. Forgetting is not an all or nothing effect where one either knows exactly where an item is or knows nothing. Instead, forgetting is the increase in the standard deviation of the probability distribution that represents knowledge of location (Fig. 1). * Correspondence: R. Biegler, Psykologisk institutt, Norges teknisk-natur- vitenskapelige universitet (NTNU), N-7491 Trondheim, Norway. E-mail address: robert.biegler@svt.ntnu.no (R. Biegler). 1 H. Tjelmeland is at the Institutt for matematiske fag, Norges teknisk-naturvi- tenskapelige universitet (NTNU), N-7491 Trondheim, Norway. 2 S. Molden is at the Centre for the Biology of Memory, Medical–Technical Research Centre, Norges teknisk-naturvitenskapelige universitet (NTNU), N-7489 Trondheim, Norway. Contents lists available at ScienceDirect Animal Behaviour journal homepage: www.elsevier.com/locate/anbehav 0003-3472/$38.00 Ó 2009 The Association for the Study of Animal Behaviour. Published by Elsevier Ltd. All rights reserved. doi:10.1016/j.anbehav.2009.06.014 Animal Behaviour 78 (2009) 819–828