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In this paper, we study N-player Colonel Blotto games with incomplete information about battlefield valuations. Such games arise in job markets, research and development, electoral competition, security analysis, and conflict resolution. For M Ï N + 1 battlefields, we identify a Bayes-Nash equilibrium in which the resource allocation to a given battlefield is strictly monotone in the valuation of that battlefield. We also explore extensions such as heterogeneous budgets, the case M È N, full-support type distributions, and network games. Revised version, February 2021

Bayesian methods are presented for categorical sampling when some observations are censored (i.e., suffer missing distinctions between categories). Such problems have been researched over the years, as they can be important in applications. However, previous work has assumed strong restrictions, such as truthful reporting, noninformative censoring, etc.Here, we attempt to remove such restrictions. In particular, we remove two of the three restrictions imposed by Dickey, Jiang and Kanade (1987). We provide Bayesian methods for cases more general than those considered by Paulino and de B. Pereira (1992, 1995), and others. Thus, it will no longer be necessary to make unrealistic assumptions commonly employed regarding the censoring model. A theorem of Identifiability-by-Conditioning is provided, allowing familiar improper prior densities. By this theorem, we obtain identical Bayesian updating results by imposing constraints on either prior, likelihood, or posterior directly. Several computational procedures are suggested, and an example is used to illustrate methods.

We wish to make inferences about the conditional probabilities p(y ( x), many of which are zero, when the distribution of X is unknown and one observes only a multinomial sample of the Y variates. To do this, fixed likelihood ratio models and quasi-incremental distributions are defined. It is shown that quasi-incremental distributions are intimately linked to decomposable graphs and that these graphs can guide us to transformations of X and Y which admit a conjugate Bayesian analysis on a reparametrization of the conditional probabilities of interest.