Deception and Unreliability: A Generalization of Extensive Form Games
by
Oliver Board
Department of Economics, Oxford University

The standard model of an extensive form game rules out two phenomena of importance in situations of dynamic strategic interaction: deception and unreliability. We show how the model can be generalized to incorporate these phenomena, and give some examples of the richer model in use.

We say that deception takes place when one player tricks another into believing that she has done something other than what she actually did. The standard model does allow moves to be uninformative (whenever information sets are non-singleton), in that it is not revealed which of several moves has been made. But they cannot be deceptive: the actual move made is never ruled out. Our extension of extensive form games relaxes the assumption that the information sets partition the set of nodes. In particular, the set of nodes believed possible after a certain move is made might not include the actual node.

Consider the story of the wooden horse in Virgil's Aeneid Book II. The Greeks have two choices, to go home and give up the war or to stay and attempt to sack Troy. The latter seems hopeless until Odysseus suggests the following plan: they should sail their ships out of sight and leave a gigantic wooden horse in front of the city. Believing the Greeks have really gone home, the Trojans accept the horse as a gift and break down their walls to wheel it into Troy. The Greeks then leap out of the horse and successfully sack the city. What we have here is a genuine case of deception: one move is made, but another is observed. Note that while standard game theory does not rule out an agent having false beliefs (for example the beliefs that justify a rationalizable strategy may well be mistaken), nothing in the structure of the game itself forces her to have these beliefs. When deception takes place, on the other hand, the agent is forced to have false beliefs. So deception is not simply a case of an agent coming to a false belief on the basis of a mistaken prior. Rather, the agent receives false information about what move has been made, and updates her beliefs according to that information.

Formally, we model deception by distinguishing the move actually made by a player, and the information (true or false) revealed by that move to every player (including the player on move herself). This second element takes the place of the information sets in the standard model.

If deception is allowed, an agent may receive two signals about which move has been made that are incompatible with each other. For example, Laocoon and Sinon give King Priam two opposing accounts of the true purpose of the wooden horse. In such cases, at least one of the signals must be unreliable. We further enrich the standard model by attaching reliability weights to each move. These weights tell us how an agent will evaluate various conflicting pieces of information. Her beliefs are updated using a modified version of the belief revision system in Board 1998 (TARK VII). The new system relaxes the AGM axiom of success, so that the latest piece of information is not always accepted. Various other sources of unreliability are also considered, including forgetfulness, where information received in the distant past is considered less reliable than more recent information.

Date received: June 10, 2000

Copyright © 2000 by the author(s). The author(s) of this document and the organizers of the conference have granted their consent to include this abstract in Atlas Conferences Inc. Document # cafi-45.