Optimal Play of the Dice Game Pig

The object of the jeopardy dice game Pig is to be the first player to reach 100 points. Each player\u27s turn consists of repeatedly rolling a die. After each roll, the player is faced with two choices: roll again, or hold (decline to roll again). If the player rolls a 1, the player scores nothing and it becomes the opponent\u27s turn. If the player rolls a number other than 1, the number is added to the player\u27s turn total and the player\u27s turn continues. If the player holds, the turn total, the sum of the rolls during the turn, is added to the player\u27s score, and it becomes the opponent\u27s turn. For such a simple dice game, one might expect a simple optimal strategy, such as in Blackjack (e.g., stand on 17 under certain circumstances, etc.). As we shall see, this simple dice game yields a much more complex and intriguing optimal policy, described here for the first time. The reader should be familiar with basic concepts and notation of probability and linear algebra

Pigtail: A Pig Addendum

The object of the jeopardy dice game Pig is to be the first player to reach 100 points. Each turn, a player repeatedly rolls a die until either a 1 is rolled or the player holds and scores the sum of the rolls (i.e., the turn total). At any time during a player’s turn, the player is faced with two choices: roll or hold. If the player rolls a 1, the player scores nothing and it becomes the opponent’s turn. If the player rolls a number other than 1, the number is added to the player’s turn total and the player’s turn continues. If the player instead chooses to hold, the turn total is added to the player’s score and it becomes the opponent’s turn. In our original article [Neller and Presser 2004], we described a means to compute optimal play for Pig. Since that time, we have also solved a number of Pig variants. In this addendum, we review the optimality equations for Pig, show how these equations change for several Pig variants, and show how the resulting optimal policies change accordingly. [excerpt

Practical Play of the Dice Game Pig

The object of the jeopardy dice game Pig is to be the first player to reach 100 points. Each turn, a player repeatedly rolls a die until either a 1 is rolled or the player holds and scores the sum of the rolls (i.e., the turn total). At any time during a player’s turn, the player is faced with two choices: roll or hold. If the player rolls a 1, the player scores nothing and it becomes the opponent’s turn. If the player rolls a number other than 1, the number is added to the player’s turn total and the player’s turn continues. If the player instead chooses to hold, the turn total is added to the player’s score and it becomes the opponent’s turn. In our original article [Neller and Presser 2004], we described a means to compute optimal play for Pig. However, optimal play is surprisingly complex and beyond human potential to memorize and apply. In this paper, we mathematically explore a more subjective question: What is the simplest human-playable policy that most closely approximates optimal play? While one cannot enumerate and search the space of all possible simple policies for Pig play, our exploration will present interesting insights and yield a surprisingly good policy that one can play by memorizing only three integers and using simple mental arithmetic. [excerpt

Referendum Design and Contingent Valuation: The NOAA Panel's No-Vote Recommendation

In 1992 the National Oceanic and Atmospheric Administration (NOAA) convened a panel of prominent social scientists to assess the reliability of natural resource damage estimates derived from contingent valuation (CV). The product of the Panel's deliberations was a report that laid out a set of recommended guidelines for CV survey design, administration, and data analysis. One of the Panel's recommendations was that CV surveys should employ a referendum approach. This method describes a choice mechanism that asks each respondent how they would vote if faced with a particular program and the prospect of paying for the program through some means, such as higher taxes. The Panel also recommended that CV referendum questions which commonly use only "for" or "against" answers should be expanded to explicitly offer an "I would-not-vote" response. The purpose of this paper is to consider the effects of such a "would-not-vote" option. In developing the test, we followed the important elements of the NOAA Panel guidelines for the design and administration of a CV survey and use what was acknowledged(by the Panel) as the most carefully developed CV questionnaire to that time, that is, the State of Alaska's study of the Exxon Valdez oil spill. Our findings suggest that when those selecting the "would-not-vote" response are treated as having voted "against" the offered program, offering the option does not alter: (a) the distribution of "for" and "against" responses, (b) the estimates of WTP derived from these choices, or (c) the construct validity of the results.

Temporal Reliability of Estimates from Contingent Valuation

In 1992 the National Oceanic and Atmospheric Administration (NOAA) convened a panel of prominent social scientists to assess the reliability of natural resource damage estimates derived from contingent valuation (CV). The product of the panel's deliberations was a report that laid out a set of recommended guidelines for CV survey design, administration, and data analysis. This paper focuses on one of these guidelines�the Panel's call for the "temporal averaging" of willingness-to-pay (WTP) responses obtained from CV surveys as one method for increasing their reliability. The panel suggested: "Time dependent measurement noise should be reduced by averaging across independently drawn samples taken at different points in time. A clear and substantial time trend in the responses would cast doubt on the 'reliability' of the finding." The purpose of this paper is to examine the temporal reliability of CV estimates. Our findings, using a CV instrument designed to measure willingness-to-pay for a program to protect Prince William Sound, Alaska from future oil spills, like the Exxon Valdez spill, exhibited no significant sensitivity to the timing of the interviews. For two samples involving independent interviews taken over two years apart, the distribution of respondents' choices "for" and "against" the protection program did not differ.

Was the NOAA Panel Correct About Contingent Valuation?

The past few years have seen a highly charged debate about whether contingent valuation (CV) surveys can provide valid economic measures of people's values for environmental resources. In an effort to appraise the validity of CV measures of economic value, a distinguished panel of social scientists, chaired by two Nobel laureates, was established by NOAA, to critically evaluate the validity of CV measures of nonuse value. The Panel provided an extensive set of guidelines for CV survey construction, administration, and analysis, and distinguished a subset of items from their guidelines for special emphasis and described them as burden of proof requirements. Of particular interest was the Panel's requirement that CV surveys demonstrate "responsiveness to the scope of the environmental insult." That demonstration has come to be called a scope test. The paper reports the findings from the first CV study that adheres to the NOAA Panel's guidelines and includes a formal scope test.

Fabrication of bismuth nanowires with a silver nanocrystal shadowmask

We fabricated bismuth (Bi) nanowires with low energy electron beam lithography using silver (Ag) nanocrystal shadowmasks and a subsequent chlorine reactive ion etching. Submicron-size metal contacts on the single Bi nanowire were successfully prepared by in situ focused ion beam metal deposition for transport measurements. The temperature dependent resistance measurements on the 50 nm wide Bi nanowires showed that the resistance increased with decreasing temperature, which is characteristic of semiconductors and insulators

Palladium-catalysed synthesis of arylnaphthoquinones as antiprotozoal and antimycobacterial agents

Malaria and tuberculosis are still among the leading causes of death in low-income countries. The 1,4-naphthoquinone (NQ) scaffold can be found in a variety of anti-infective agents. Herein, we report an optimised, high yield process for the preparation of various 2-arylnaphthoquinones by a palladium-catalysed Suzuki reaction. All synthesised compounds were evaluated for their in-vitro antiprotozoal and antimycobacterial activity. Antiprotozoal activity was assessed against Plasmodium falciparum (P.f.) NF54 and Trypanosoma brucei rhodesiense (T.b.r.) STIB900, and antimycobacterial activity against Mycobacterium smegmatis (M.s.) mc(2) 155. Substitution with pyridine and pyrimidine rings significantly increased antiplasmodial potency of our compounds. The 2-aryl-NQs exhibited trypanocidal activity in the nM range with a very favourable selectivity profile. (Pseudo)halogenated aryl-NQs were found to have a pronounced effect indicating inhibition of mycobacterial efflux pumps. Cytotoxicity of all compounds towards L6 cells was evaluated and the respective selectivity indices (SI) were calculated. In addition, the physicochemical parameters of the synthesised compounds were discussed

MXene: a promising transition metal carbide anode for lithium-ion batteries

Herein we report on Li insertion into a new two-dimensional (2-D) layered Ti₂C-based material (MXene) with an oxidized surface, formed by etching Al from Ti₂AlC in HF at room temperature. Nitrogen sorption of treated powders showed desorption hysteresis consistent with the presence of slit-like pores. At 23 m² g-¹, the specific surface area was an order of magnitude higher than untreated Ti₂AlC. Cyclic voltammetry exhibited lithiation and delithiation peaks at 1.6 V and 2 V vs. Li+/Li, respectively. At C/25, the steady state capacity was 225 mAh g-¹; at 1C, it was 110 mAh g-¹ after 80 cycles; at 3C, it was 80 mAh g-¹ after 120 cycles; at 10C, it was 70 mAh g-¹ after 200 cycles. Since Ti₂C is a member of the MXene family - where M is an early transition metal and X is C and/or N - that to date includes Ti₃C₂, Ta₄C₃, TiNbC, and (V₀.₅,Cr₀.₅)₃C₂, our results suggest that MXenes are promising as anode materials for Li-ion batteries

Polyacrylonitrile-containing amphiphilic block copolymers: self-assembly and porous membrane formation

The development of hierarchically porous block copolymer (BCP) membranes via the application of the self-assembly and non-solvent induced phase separation (SNIPS) process is one important achievement in BCP science in the last decades. In this work, we present the synthesis of polyacrylonitrile-containing amphiphilic BCPs and their unique microphase separation capability, as well as their applicability for the SNIPS process leading to isoporous integral asymmetric membranes. Poly(styrene-co-acrylonitrile)-b-poly(2-hydroxyethyl methacrylate)s (PSAN-b-PHEMA) are synthesized via a two-step atom transfer radical polymerization (ATRP) procedure rendering PSAN copolymers and BCPs with overall molar masses of up to 82 kDa while maintaining low dispersity index values in the range of Đ = 1.13-1.25. The polymers are characterized using size-exclusion chromatography (SEC) and NMR spectroscopy. Self-assembly capabilities in the bulk state are examined using transmission electron microscopy (TEM) and small-angle X-ray scattering (SAXS) measurements. The fabrication of isoporous integral asymmetric membranes is investigated, and membranes are examined by scanning electron microscopy (SEM). The introduction of acrylonitrile moieties within the membrane matrix could improve the membranes’ mechanical properties, which was confirmed by nanomechanical analysis using atomic force microscopy (AFM)