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We show that the playing sequence--the order in which players update their actions--is a crucial determinant of whether the best-response dynamic converges to a Nash equilibrium. Specifically, we analyze the probability that the best-response dynamic converges to a pure Nash equilibrium in random $n$-player $m$-action games under three distinct playing sequences: clockwork sequences (players take turns according to a fixed cyclic order), random sequences, and simultaneous updating by all players. We analytically characterize the convergence properties of the clockwork sequence best-response dynamic. Our key asymptotic result is that this dynamic almost never converges to a pure Nash equilibrium when $n$ and $m$ are large. By contrast, the random sequence best-response dynamic converges almost always to a pure Nash equilibrium when one exists and $n$ and $m$ are large. The clockwork best-response dynamic deserves particular attention: we show through simulation that, compared to random or simultaneous updating, its convergence properties are closest to those exhibited by three popular learning rules that have been calibrated to human game-playing in experiments (reinforcement learning, fictitious play, and replicator dynamics).

Certain Lathyrus species have recently been found suitable as multipurpose legume crops in the southern Australian dryland regions, with a potential growing area of 100,000–300,000 ha. However, their use is limited by the presence, mainly in seeds, of 3-(- N -oxalyl)- l -2,3-diamino propionic acid (ODAP), an agent which causes lathyrism. Researchers at the Centre for Legumes in Mediterranean Agriculture (CLIMA) have selected a high yielding variety of Lathyrus cicera (cv. Chalus) which contains consistently low levels of ODAP. The aim of the present study was to assess the nutritional value and animal health aspects of L. cicera cv. Chalus grain as a feed for sheep. Comparisons were made against lupin grain ( Lupinus angustifolius ). Eighty individually penned Merino wethers (35±0.3 kg) were allocated to four treatment groups of 20 sheep. Following a 4-week adjustment period, sheep were fed ad libitum the following treatment diets for up to 13 weeks: 35% lathyrus, 35% lupin, 70% lathyrus and 70% lupin. The remainder of the diet was oaten hay and minerals. After 10 weeks on treatments, 14 sheep per group were killed for carcass assessment. The remaining six sheep were fed to maintenance ME requirements for 1 week followed by a 10-day faecal and urinary collection period. Analysis of the grains showed that lathyrus contained less protein (28% versus 36%), fat (0.7 versus 4.1%) and fibre (25% versus 36% NDF), but more starch (42% versus 1%) and anti-nutritional compounds (ODAP, tannins, trypsin inhibitor) than lupin. Essential amino acid composition was similar for the two grains, as were in sacco degradabilities of dry matter (DM) (84% versus 81%) and protein (92% versus 94%). The feeding experiment showed that lathyrus had a higher nutritional value than lupin in terms of voluntary feed intake, liveweight gain ( P P P =0.05). Wool growth reflected ME intake, and there was no independent effect of grain type ( P >0.05). Results from the balance study showed that ME concentration was the same for both grains (14 MJ/kg DM). Microbial crude protein (CP) synthesis, estimated from urinary purine derivatives, was significantly lower for lathyrus than lupin, and was lower at 70% grain inclusion than at 35% ( P There were no visible or biochemical signs of ill health (in-appetence, lethargy, shaking or instability) in any sheep fed lathyrus. Several sheep fed the 70% lupin diet had mild diarrhoea, and two were eventually removed from the experiment due to anorexia. Meat from sheep fed lupin tended to be yellower than that from those fed lathyrus ( P =0.05). Apart from this, there were no differences in meat quality due to grain type (e.g. redness, pH, taste or tenderness). In conclusion, compared with lupin grain, low ODAP L. cicera grain appears to be of high nutritional value for sheep, with no evidence of adverse effects on sheep health.

Fluoro chlorides are major products in the reaction of methyl hypochlorite with certain olefins in the presence of boron trifluoride.

SEVERAL investigators have shown that a chick’s viability and its storage of the vitamins at hatching time are influenced by the intake of these specific essential nutrients by the parent stock. It is well established that some other as yet unidentified factor (or factors) is transmitted from the hen to the egg and subsequently to the newly hatched chick. When placed on a diet deficient in this constituent, the chick’s ability to grow at a normal rate is dependent upon an adequate storage of this unknown. Bird and associates (1946) found that the viability of chicks was low when hens were fed a soybean oil meal supplemented diet, adequate in all known factors. Mortality of chicks from hens receiving a soybean oil meal diet was 38.0 percent, whereas, hens receiving this diet plus 5.0 percent cow manure or sardine meal produced chicks with a mortality rate of 12.0 and 11.0 . . .

PEROSIS, an anatomical deformity of the leg bones of young chicks, was first described by Hunter and Funk (1930). The condition is associated with a high intake of calcium and phosphorus, and a low intake of manganese (Hunter et al., 1931; Milby, 1934; Wilgus et al., 1936–1937; Clifcorn et al., 1938). Later, the work of Wiese et al. (1939) showed that the blood and bone phosphatase activity of perotic birds was lower than that of chicks receiving an adequate amount of manganese in their diet. As the leg bones of chicks with perosis are bent and twisted from their normal shape it was thought that there might be an abnormal distribution of the enzyme phosphatase in the bones of these chicks. In this paper we wish to report the results we have observed on the distribution of this enzyme in the leg bones of perotic and normal chicks. EXPERIMENTAL

VITAMIN B12 is recognized as an essential dietary nutrient for hatchability. Limited data is now available concerning animal protein factor supplements as a source of this vitamin and other factors. Carver and McGinnis (1950a) found that a fermentation APF supplement was effective at levels of 0.5, 0.25, and 0.125 percent in improving hatchability from a low level or maintaining high hatchability in a nine-week experiment with hens on litter or wire. Another APF supplement with a content of 12.5 mg. of vitamin B12 per pound was reported by Hill et al. (1950) to be as effective in maintaining hatchability as 6 percent fish meal. These experimental birds were reared from day-old to 10 months on wire floors and fed the same supplements during the entire period. This product, when used at 0.11 percent, apparently contained adequate B12 to support good hatchability and permitted even greater transfer to the chick than . . .

SEVERAL investigators have reported that vitamin B12 is essential for hatchability, and that it can successfully replace animal proteins as a source of vitamin B12. A.P.F. supplements containing this vitamin improved hatchability with an all-vegetable diet as reported by Lindstrom et al. (1949a), Carver and McGinnis (1950a), and Hill et al. (1950). Crystalline vitamin B12 has also been shown to support good hatchability by Petersen et al. 1950), Carver and McGinnis (1950b), and Milligan and Combs (1950). This paper reports the results obtained when hens, which had received various levels of vitamin B12 for an 8-week experimental period, were returned to the all-vegetable basal ration. The value of feeding antibiotics on hatchability with or without vitamin B12 has been investigated and the results are presented. EXPERIMENTAL Ninety-six White Leghorn hens were placed in individual raised wire-bottom laying cages at approximately 5 months of age. A ration low in vitamin B12, . . .

THE production of safflower has continued to increase in this country, resulting in renewed interest in the use of the by-product of this seed as an additional source of protein for poultry feeding. Undecorticated safflower meal, because of the heavy and fibrous hull, contains only 20 percent protein and as high as 35 percent fiber. By a decortication process it is possible to remove a large portion of the hull, which will result in the production of a meal in which the protein may exceed 50 percent with a resulting low fiber content. Most meals produced by this process generally contain from 35 to 40 percent protein and 12 to 15 percent fiber. A study of safflower protein for chick growth was first reported by Kratzer and Williams (1947). Two lots of meal, prepared in their laboratory, contained 29.1 and 27.2 percent crude protein. The meal was fed as the …

VARIOUS workers have shown that fish meal and other animal products contain an unidentified hatchability factor(s) that is absent in plant products. Lampman and associates (1944) have shown that rations supplemented only with plant proteins, such as soybean oil meal, resulted in reduced hatchability. Rubin, et al., (1946) showed that an unidentified chick growth factor present in cow manure was also found in the droppings of hens. Kennard and co-workers (1948) reported experimental work which indicated that the unidentified hatchability factor was present in hen feces, especially when a built-up litter program was used. Since McGinnis and co-workers (1947) presented evidence that the synthesis of this factor occurs after feces have been voided, it would appear that it was obtained by the hens through coprophagy. Rubin, Groschke, and Bird (1947) reported that an acid precipitate of a water extract of cow manure was highly potent in the hatchability factor required . . .

THE requirements for a number of the known vitamins have been determined for the laying hen. These requirements are for vitamin A and D, riboflavin, pantothenic acid, folacin, and pyridoxine and have been compiled by the National Research Council (1954) as averages of results of several comparable experiments. Using high energy rations, Thayer (1953) reported that levels of niacin, riboflavin, pantothenic acid, and folic acid above recommended requirements were necessary for maximum egg production and feed efficiency. However, Berg and Bearse (1956), using high energy rations, did not obtain improved egg production and feed efficiency with an additional B-complex vitamin supplement. These workers did report significant differences in body weight gain in favor of the birds fed the vitamin-supplemented rations. This concurred with the results of Thayer. The increased feed efficiency obtained when high energy rations are fed to growing chickens has prompted considerable study of the effect of high . . .

WORK leading to the development of a ration for the uniform production of perosis (slipped tendon) in chicks was presented in the previous paper. In this paper we wish to give the results obtained from studies on factors which prevent the development of this syndrome. Jukes (1937) has recently reviewed the meager work available on the perosis-preventing properties of certain feeds. EXPERIMENTAL Day-old White Leghorn chicks were divided into groups of five chicks each and placed in suitable cages. In the majority of the work ration 604 was used as the basal ration, but results obtained with other closely related rations are also included. The supplements tested were added at the expense of the whole ration. The results are tabulated in Table 1. It is readily evident that the addition of 15 percent of rice bran gave complete protection in all trials. The result checks quite well with the conclusion . . .

VITAMIN B12 has been shown to support good hatchability in all-vegetable rations, containing soybean oil meal as the protein supplement, by Petersen et al. (1950), Carver and McGinnis (1950), Milligan and Combs (1950), Peeler et al. (1951), Johnson (1951), Yacowitz et al. (1952), and Milligan et al. (1952). Some conflicting reports have appeared in which B12 alone did not maintain hatchability at a high level. Olcese and Couch (1950), using a more purified diet containing sucrose and soybean protein, found that vitamin B12 would not support hatchability after the eighth week of feeding. Peeler et al. (1951) reported that hens kept in laying cages for a period of more than one year showed reduced hatchability due to early embryonic mortality which could not be improved by weekly injections of vitamin B12. Vitamin B12 supported good hatchability of those embryos which passed this early embryonic period. The purposes of the experiments …

The use of high-energy low-fiber broiler rations containing large amounts of animal proteins has come at a time when animal products for livestock feeding are extremely scarce and costly. Almost simultaneously animal protein factor (B12) preparations have become available as supplements for poultry rations. Crystalline vitamin B12 was shown by Ott, et al. (1948) to give response similar to the animal protein factor. Lindstrom, et al. (1949) found that a vitamin B12 concentrate gave approximately the same response for chick growth as condensed fish solubles. It had previously been shown by Petersen and Lampman (1949) that the use of herring fish meal at a level of 10 percent in a high-energy broiler ration permitted the omission of liver meal, choline and niacin. It was desired to determine how effective an animal protein factor concentrate * would be in supplementing an all-cereal ration containing soybean oil meal. PROCEDURE Four duplicate lots of . . .

Animal protein supplements including liver meal, condensed fish solubles, fish meal, meat meals, and milk products have been shown to contain an unknown factor (or factors) required by chickens for reproduction and early growth. Rubin and Bird (1946) and Whitson, Titus and Bird (1946) found dried cow manure to contain similar factors. An animal protein factor supplement, available following the isolation of vitamin B12, was found by Lindstrom et al. (1949a) to contain factors essential for hatchability when hens received an all-plant ration. Lillie, Olsen and Bird (1949) have reported that crystalline vitamin B12 injected into eggs improved hatchability when the breeder hens were fed a diet deficient in this vitamin. Subsequent chick studies also showed improved livability and feathering following the B12 injections into the eggs. Their data is not completely in agreement or conclusive, primarily because of fairly high hatchability of eggs from the basal diet without vitamin . . .

Crystalline vitamin B12, isolated from liver by Rickes et al. (1948) was shown by Ott et al. (1948) to give response similar to the animal protein factor (APF ) required for chick growth. Preparations of a microbiologically produced material by Stokstad et al. (1948) have also been shown to be highly active as a source of APF, when measured by chick assay. A vitamin B12 ** concentrate (4.4 ug. of B12 per gram) was fed at a 1% level to APF depleted chicks to determine its effect upon chick growth and to APF depleted laying hens to determine the effect upon hatchability. The concentrate was used to supplement both a synthetic and a natural type chick diet. The synthetic diet contained alpha soybean protein, cerelose, soybean oil, salts, amino acids, and vitamins A, D, and B-complex supplements. The natural type diet contained wheat bran, ground corn, oats and wheat, soybean oil . . .