How deep does a chess Grandmaster think? This question has been asked many times, and yet there is hardly a definite answer. Raw depth and pure calculation are certainly not the only factors in the thinking process of a chess player, but it would be interesting to know more about the relationship between search depth and playing strength, so that the strength of a given player (which is usually expressed in terms of an Elo rating) can be said to correspond to a certain equivalent depth (of some

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... given engine). Since the thinking depth of a human player is difficult to determine, we carry out an experiment with a chess engine running at different search depths in order to obtain an average score that can be translated into a rating difference in the Elo scale. However, knowing the rating difference is not sufficient; we need have at least one value of engine depth for which the corresponding Elo rating has been estimated, so that the Elo ratings for other values of search depth can also be determined. In order to obtain the Elo rating that corresponds to HOUDINI 1.5a 64-bit running at a fixed iteration depth of 20 plies, we carry out an analysis of the quality of play at the Candidates Tournament 2013. From these results, we show how the search depth of that particular engine correlates with the Elo ratings of human players. The paper also discusses related work on self-play experiments and the effect of diminishing returns, which becomes apparent in our experiment. The Impact of Search Depth on Chess Playing Strength June 2013 chess engine working at some fixed iteration depth d. Here, as in Heinz (2000), "fixed iteration depth" refers to imposing a depth limit on the iterative deepening of a chess program. By equating the strength of a player to the search depth of a chess engine, the underlying assumption is that the stronger a player is, the higher will be the equivalent depth d. This does not mean that the player will follow the same iterative deepening procedure up to depth d, as the chess program will do, but only that the player is on the same level of strength as a chess engine with fixed iteration depth d. This means that, over a large number of games, the average score between player and engine would be close to 0.5 (i.e., closer to 0.5 than what would happen for d−1 or d+1). The main goal of this work is to establish a relationship between player strength and equivalent depth, so that it is possible to determine how deep the engine should search in order to reach a given playing strength. Even though the search depth of an engine does not necessarily relate to the thinking process of a human player, the relationship between search depth and playing strength will nevertheless provide a feeling for how deep a Grandmaster thinks in comparison to other less skilled players.