After graduating from the Heidelberg University Faculty of Law in 1873, he served in the Russian Army as a lieutenant-general.
1873 he defeated prince Villafranca (+7 -2 =1)
1874 he beat W. Liselle (+5 -1 =1); Polner (+1 -0 =) and Schoumoff (1-0)
1880 he played a series of games with Serafino Dubois, with even results.
1882 he defeated M.A. Clerc in a series of 5 games.
1888 he beat M.A. de Smitten (+7-2=3)
1901 at Monte Carlo, Dadian supplied the 500 franc brilliancy prize.
1903 at Monte Carlo, Dadian was President of the organizing committee.
1904 at Monte Carlo, Dadian supplied 2 additional prizes of 300 francs.
1905 Dadian opened the Chess Congress in Barmen.

Besides the tournament in Homberg. Dadian is said to have participated in  tournaments at Rome, to have won the Petersburg amateur chess tournament in 1881-1882 as well as placing second in the  Kiev city tournament, behind  Nikolaev in 1900  and first in the Kiev city tournament in 1904.

Dadian died in Kiev on June 12, 1910.

From the November 1910 BCM (copied from Edward Winter's Chess Notes #1490)

We learn with regret from La Stratégie that Prince Andre Dadian, of Mongrelia [sic], died at Kieff, on 12^th June last. He was well known throughout the chess world as a gifted player, and a munificent patron of the game.
 

Note 1 - Prince Andrei  Dadiani is often cited as having been the Russian candidate for the Bulgarian crown in 1887. While it's true that a Prince of Mingrelia was such, it was Prince Nikolai, not Andrei.
Note 2 - in the photograph, Prince Dadian seems to have a full beard. It's more likely that he is wearing a high fur collar, giving the illusion of a full beard.
Note 3 - much of the information on Dadian's life was extracted from a contribution to Edward Winter's Chess Notes by Bob Meadley.
Note 4 -   Earliest found reference to Dadian - from Staunton's chess column, The Illustrated London News,   September 14, 1867. pg.289.

CHESS AT BADEN-BADEN.--- The winner of the Emperor's prize, Mr. Kolisch, gave a séance d'echecs, at Baden-Baden, on the 3rd inst. This séance was given at the invitation of M. Benazet, the celebrated Fermier des jeux; and, besides her Majesty the Queen of Prussia, who entered for a few moments, was honoured by the presence of their Highnesses Prince Stourdra and Monseigneur Mustapha Pacha, the Viceroy of Egypt's brother; Prince Bibesco; the Prince of Mingrelia; Baron de Krudner, the grandson of the famous Mdme. de Krudner, the foundress of La Sainte Alliance; Baron Ziegesar, and many others of lesser note. Princess Souwaroff remained during the whole sitting, and played herself a game of singular brilliancy with Baron Haber. Mr. Kolisch played seven games against as many different adversaries, two consultation parties, and one game without seeing the board,  -- in all, ten games simultaneously ; and he won them all.

A translation of the Dadiani entry in the  Russian Encyclopedia of Chess
(Chief Editor: A. Karpov -  published in Moscow in 1990.):

DADIANI Andrei Davidovich (1850, city of Zugdidi,- 13.6.1910, Kiev), first Georgian chess player, achieved renown in Russia and Europe; prince, general-lieutenant of the Russian army; chess Maecenus. Participant in tournaments of amateur-chessplayers at St. Petersburg (1881-82; 1st) and Kiev (1900; 2nd), friendly matches with A. Solovtsov (1878), S.Dubois (1879), G. Gelbak (1895). Met across the board with I. Shumov (1880), had friendly relations with I.S. Turgenev. Took part in the organizational committees of the international tournaments: Monte-Carlo (1903/1904), Milan (1906) and others.
~[Shiffers E.], Fins de partie de S.A.S. le Prince Dadian de Mingrelie, Kiew, 1903; Giorgadze T., Ygraet A. Dadian, Tbilisi, 1972 (in Georgian).

Note 1: (from the catalogue at the Cleveland Public Library):
Author: Dadian, Andrei Davidovich, 1850-1910.
Title : Fins de partie.
Publisher : Kiew, Impr. T. Bencionowski  [1903]
Description : 215 p. incl. plates, port.
Notes : Flex Key : o07454240
Subject Heading(s) : Chess--End games

Note 1-A:
The Schiffers book is in French and was published in 1903. Here is how Panov describes it (from pgs. 293-294 of his book on Tschigorin, "The Pauper Knight"):

...Moreover, in the year 1903, by the financing of the Prince himself there appeared in Kiev in the French language (Russian, evidently, in his opinion did not suit such a noble personage as himself) the collection "The Chess Endings of His Highness Prince Dadian of Mingrelia" It was a luxuriously designed book, containing a hundred endings, shown in colored diagrams. In all of the endings, of course, the much-esteemed self-publisher invariably and strikingly "won".

The foreword to the collection and the annotations were written by Schiffers for the appropriate publicity fee. Also included were chess problems by Alapin and the German problemist Bayer, dedicated to the Prince.

The endings, almost all of which were played against players unknown in the chess world , finished with simple "typical" combinations known to any category of chess player, that could only impress such an ignorant player as the Prince himself. Many of them were not in actual fact played but composed by the Prince. Exactly why Dadian came out with a collection of "endings", as opposed to games (as is customary), was because when one scrutinizes a game it is already easy to judge the skill of the player from the opening and from the middle-game and also from how the final combination organically flows from the previous play. To forge a game, as we’ll see further on, is extraordinarily hard, but an ending is much easier.
 

Note 2 (by WilhelmThe2nd):
The Kiev, 1900 tournament was actually the first Kiev championship which was probably sponsored by Dadian. He came second to Nikolaev. (This info is from Tschigorin's chess column in Novoe Vremya of 13th (26th) Dec. 1906) Something must have soured between Dadian and the Kiev chess club because in the years after 1900 there are a few announcements in the British Chess Magazine where it says they have received a letter from Dadian saying he is not a member of the club and does not even know where it is located in Kiev.(!) In Panov's historical novel about Tschigorin The Pauper Knight (/The Poor Knight) he says it came about because a chess column in a local paper run by the Kiev chess club published one of the Princes losses. The Prince was so infuriated he challenged the editor of the column to a duel. I have seen no substantiation of this claim though.

PRINCE DADIAN, OF MINGRELIA

We wonder how many of our readers know where Mingrelia is. It is a province of Asiatic Russia, lying between the Caucasus and the Black Sea. It has, we believe, a language of its own, and the Dadian or Prince, we understand, is the titular ruler. Obscure though his country may be, the name of the Prince is known throughout the chess world as that of an original and brilliant player. He has been good enough to send us a selection of his games, which from time to time, as space permits, we shall have much pleasure in publishing. We have also received from a correspondent some account of his Most Serene Highness' life, from which we gather the following particulars. He was born at Zondidi, the capital of Mingrelia, and from infancy displayed an extraordinary liking for various branches of science. His family used to spend the winters at Paris, and from the age of 14 the Prince began to write good French verse, and as a exercise to compose fictitious tales which astonished his Professors. Endowed with a powerful memory, he can recite an immense quantity of poetry; he speaks six modern language, and his erudition is known throughout Europe. He has played chess from his boyhood, and at Vienna, in 1882, after the banquet which took place at the close of the International Tourney there, he played a blindfold game with such accuracy that the masters who were present applauded him heartily.
Nevertheless, on account of his other occupations the Prince rarely plays chess, and is far from having the practice of the other masters. His end games are very beautiful, and will remain as chef d'oeuvres in the literature of chess. The following games will be found worthy specimens of his skill.
[There follows the Prince's games vs. Col. Veriguine and M. Bitcham with notes by C. E. Rankin]
 

from Steinitz' chess column in the New York  Daily Tribune Sunday, June 5, 1892 ---

Mr. Steinitz has received from Messrs. Knauth, Nachod & Kuhne, bankers, of South William St., New York, the sum of 1,000 francs by cable order from Odessa of his Serene Highness Prince Dadian of Mingrelia. Further explanations will no doubt arrive in due course by mail, and in the meanwhile it is surmised that the generous donor intends to award to Mr. Steinitz that amount as a prize for the most brilliant game of the last championship match against Mr. Tschigorin at Havana-probably the fourth game, a Ruy Lopez, in which much was effected after sacrificing two Rooks -- in the same manner as was done by His Highness in the match between the same players in 1889, on which occasion the brilliancy prize also fell to Mr. Steinitz. But it is also likely that the donation is a supplementary subscription to the Jubilee Testimonial Fund, which was organized last year in celebration of Mr. Steinitz having held the championship of the world for twenty-five years, and that His Highness had no timely notice of the subscription, owing to his frequent travels in distant parts of the Caucasus in the execution of his military duties. In any case the handsome present will be gratefully appreciated by Mr. Steinitz, and will also no doubt create a gratifying impression among all lovers of the game.

from Steinitz' chess column in the New York  Daily Tribune Sunday, June 26, 1892 ---

A letter from Prince Dadian of Mingrelia confirms that the sum of 1,000 francs, which, as announced in our issue of June 5, His Serene Highness had forwarded by cable to Mr. Steinitz, is intended as a supplementary subscription to the Steinitz Jubilee Testimonial Fund. His Highness is stationed at Zougdidi, in the Caucasus.

Note 3 (by WilhelmThe2nd):
In his book on Tschigorin, Panov jokes about Dadian`s letters to chess magazines being accompanied by large checks. Perhaps this explains why, for instance, the French magazine,  La Stratégie, published so many of Dadian`s games over the years.

 

Note 4 (by WilhelmThe2nd):
Some of it does give an idea of his character but I think his play has much to do with his era and his background. The Old School (Morphy-Anderssen) vs. Modern School (Steinitz) debate was going on during his mature years and obviously Dadian preferred the Morphy-Anderssen style. He came from a military background where bravery, daring and courage were highly valued so it is not surprising that he delighted in direct, swashbuckling sacrificial attacks. There is a story that he played a consultation game with Schiffers on his side against two of the participants in the 1903 Russian Ch. which was held in his hometown of Kiev. Before move ten Dadian insisted to Schiffers that they play a crazy, unsound sacrifice and Schiffers was so annoyed that he got up and left the game. (Schiffers did publish the game in his Niva column but with only one comment on the name of the opening.) Another factor might be that he was not the oldest son in his family so his brother would have received much attention (I think his brother was suggested as the monarch of Bulgaria at one point. BCM confused Prince Andre with his brother and mentioned this news noting he was a brilliant chess player.) Perhaps his desire to shine at chess was a way of getting attention for himself. Panov in his book on Tschigorin is pretty scathing about Prince Dadian. Since Panov was writing from the Marxist viewpoint he thought the Prince was an idle, vain narcissist who was just another member of the exploiting classes. I am interested to know why Dadian seemed to lose interest in chess after 1904 or so. You don't really see anything about him in the chess literature after that until he died in 1910. I haven't been able to trace any contemporary Russian obituaries, if there were any.