When giving an nalysis of knowledge in multiagent systems, one needs a framework in which higher-order information and its dynamics can both be represented. A recent tradition stoxting in origina work by Plaza treats all of knowledge, higher-order knowledge, and its dynamics on the sae foot. Our work is in that tradition. It also fits in approaches that not only dynaize the epistemics, but also epistemize the dynamics: the ac- tions that (groups of) agents perform oxe epistemic actions. Different agents may have different information about which action is taking place, including higher-order information. We demonstrate that such information changes require subtle descriptions. Our contribution is to provide a complete axiomatization for n action language of vn Ditmoxsch, where an action is interpreted as a relation between epistemic states (pointed models) and sets of epistemic states. The applicability of the framework is found in every context where multiagent strategic decision making is at stake, and aready demonstrated in gae-like scenoxios such as Cluedo and coxd games.

this article. Iwant to thank the following people for their contributions: MikeAtkinson, Ben Handley, Wiebe van der Hoek, Gerard van Kempen, Lambrecht Kok, Barteld Kooi, Jarda Opatrny, Rohit Parikh, Marc Pauly, Alexander Shen, B. Vorselaars, RinekeVerbrugge. Marc Pauly originally brought the Russian cards problem to my attention. Lambrecht Kok came up with the `modulo seven' solution at the time I didn't yet know of the Russian Olympiad solution. MikeAtkinson came up with the `projective plane' solution. (Its equivalent in conjunctive normal form is myown.) Gerard van Kempen en B. Vorselaars both came up with the `sum is 12' solution in the prize competition of the journal Natuur & Techniek. Alexander Shen provided valuable comments on my motivation of the common knowledge requirements. Proposition 5 is by Ben Handley. Jarda Opatrny pointed out the validity of the six hand solutions. Wiebe van der Hoek's intensive reading was invaluable at the stage of nishing this article

Pit is a multi-player card game that simulates the commodities trading market, and where actions consist of bidding and of swapping cards. We define a simplification of that game for which we present a detailed description of all dynamical game features. The description is in a standard language for dynamic epistemics. This formalization is then used to outline the game theory for a simplification of the Pit game. This uncovers some interesting equilibria.

this article is based will appear later this year (see Kooi (2003))

Introduction Imagine three players Anne, Bill, and Cath, each holding one card from a `stack' of three (known) cards clubs, hearts, and spades, such that they only know their own card but do not know which other card is held by which other player. Assume that the actual deal is that Anne holds clubs, Bill holds hearts and Cath holds spades. Now Anne announces that she does not have hearts. What was known before this announcement, and how does this knowledge change as a result of that action? Before, Cath did not know that Anne holds clubs, but afterwards she knows that Anne holds clubs. This is because Cath can reason as follows: "I have spades, so Anne must have clubs or hearts. If she says that she does not have hearts, she must therefore have clubs." Bill knows that Cath now knows Anne's card, even though he does not know himself what Anne's card is. Both before and after, players know which card they hold in their hands. Note that the only change that appears to have taken place i

We give some semantic results for an epistemic logic incorporating dynamic operators to describe information changing events. Such events include epistemic changes, where agents become more informed about the non-changing state of the world, and ontic changes, wherein the world changes. The events are executed in information states that are modeled as pointed Kripke models. Our contribution consists of three semantic results. (i) Given two information states, there is an event transforming one into the other. The linguistic correspondent to this is that every consistent formula can be made true in every information state by the execution of an event. (ii) A more technical result is that: every event corresponds to an event in which the postconditions formalizing ontic change are assignments to ‘true ’ and ‘false ’ only (instead of assignments to arbitrary formulas in the logical language). ‘Corresponds ’ means that execution of either event in a given information state results in bisimilar information states. (iii) The third, also technical, result is that every event corresponds to a sequence of events wherein all postconditions are assignments of a single atom only (instead of simultaneous assignments of more than one atom). 1

n de kennislogica. 1 Schijnbare oplossingen De voorwaarden voor de oplossing van het probleem zijn dat: # Anna kent Berts kaarten, (######) # Bert kent Anna's kaarten, (######) # Cees kent geen van Anna's of Berts kaarten. (#########) Neem nu verder aan dat de hand van Anna #0# 1# 2# is (schrijf 012), dat de hand van Bert 345 is, en dat Cees kaart 6 heeft. De `foute oplossing van de Olympiade' genoemd in [12] is: ######## ######## ########## ## ###### ### ######## hans@cs.otago.ac.nz ### #### ### ####### #### ## ### ##### #### ### ################# 1 Anna zegt: \Als jij 0 niet hebt, dan heb ik 012." en Bert zegt: \Als jij 3 niet hebt, dan heb ik 345." (#) Waarom `lijkt' dit een oplossing? Stelt u zich het standpunt voor van een `insider' Dirk, die in ieders kaarten kan kijken: Dirk zegt: \Als Bert 0 niet heeft, dan heeft Anna 012,enalsAnna 3 niet heeft dan heeft Bert 345." (##) We bereiken dan een informatietoestand waarin ######, ###### en ######### allemaal het geval

knowledge, chance, and change proefschrift ter verkrijging van het doctoraat in de Wiskunde en Natuurwetenschappen aan de Rijksuniversiteit Groningen op gezag van de Rector Magnificus, dr. F. Zwarts, in het openbaar te verdedigen op