Initial setups for Stratego programs

Transcription

1 Summary What setups give best chances to win and are playable for Stratego programs? Where can answers to these questions be found? Information about initial setups is available on internet and in literature. Authors give fragmentary advices. But the advices are far from complete and most advices lack the level of detail that may enable the programming of a function for the automatic creation of playable setups. Statistical research of the Gravon database offers a means to verify rules mentioned on internet and in literature. The analysis has proven that most advices should be rejected or should be applied with considerable nuance. Only a minor part of these advices appears to be trustworthy, profitable and suitable for the implementation in a Stratego program. The Gravon database appears to be not the ultimate oracle. Its volume allows only conclusions about the distribution of top ranks on a global level of detail. It is not suitable for the verification of rules on a deeper level of detail. That leaves many questions unanswered. So there is a lot of room for human expertise and imagination for anybody who wants to make a program that generates playable setups for a Stratego program. 0

2 Contents Introduction Problem definition The approach The Gravon database... 7 Advices for setups and their validity Literature and websites Classification of advices The distribution of marshal, general, spy and flag The protection of the flag Blockages, mobility and protection....6 The placement of scouts The placement of miners Cooperative units Advices without an analysis Conclusion: advices do not constitute a base for a Stratego program What to do when advices are not sufficient?... 0 Basic concepts.... Lanes.... Distances and domains.... Front and rear..... Three separate initial strategies in the front parts of the domains..... One common final strategy....4 Simple rank statistics of squares, rows and columns Row statistics of the flag Relevant criteria and stepwise refinement The distribution of top ranks over the board Marshal and general together or apart Marshal and general front or back Marshal, general and flag Marshal, general, flag and front versus rear Detailed views of marshal, general and flag Symmetrical configurations Mirror configurations with marshal and general in the same domain Mirror configurations with marshal and general in different domains Setup against setup... 4 Author: Han Wolf

3 4.6 Summary and conclusion The distribution of sub-top ranks over the board The influence of colonels Rules for marshal and general in the same domain Rules for the marshal and general in different domains Significant cases on a deeper level of detail The influence of the spy Configurations ordered by frequency Configurations ordered by score Additional remarks Protection of the flag by bombs Flag bomb configurations ordered by frequency Flag bomb configurations ordered by score Flag bomb configurations combined with marshal general configurations Most used marshal-general-flag-bomb configurations Best performing marshal-general-flag-bomb configurations Protection of the flag by mobile ranks Protection by high ranks Active protection by high ranks plus passive protection by bombs The flag in the corner The flag at an edge The flag in the middle The distribution of lower ranks over the board The team concept Lower ranks in the rear Lower ranks in a front domain without a top rank Statistics for lower ranks in teams The choice of a strategy and suitable setups for a Stratego program A suitable strategy for a Stratego program Strategy for setups with marshal and general in the same domain Strategy for setups with marshal and general in different domains Teams that fit to an offensive strategy Gambling attacks and lane control Gambling attacks will occur Chances with gambling attacks Domain control by top ranks Author: Han Wolf

4 9.5.4 Defence of domains without top ranks Consequences for setups in Stratego programs Summary and conclusions APPENDIX A Basic rank statistics... 0 A. Marshal... 0 A.. Squares... 0 A.. Rows... A.. Columns... A. General... A.. Squares... A.. Rows... 4 A.. Columns... 5 A. Colonel... 6 A.. Squares... 6 A.. Rows... 7 A.. Columns... 8 A.4 Major... 9 A.4. Squares... 9 A.4. Rows... 0 A.4. Columns... A.5 Captain... A.5. Squares... A.5. Rows... A.5. Columns... 4 A.6 Lieutenant... 5 A.6. Squares... 5 A.6. Rows... 6 A.6. Columns... 7 A.7 Sergeant... 8 A.7. Squares... 8 A.7. Rows... 9 A.7. Columns... 0 A.8 Miner... A.8. Squares... A.8. Rows... A.8. Columns... Author: Han Wolf

5 A.9 Scout... 4 A.9. Squares... 4 A.9. Rows... 5 A.9. Columns... 6 A.0 Spy... 7 A.0. Squares... 7 A.0. Rows... 8 A.0. Columns... 9 A. Bomb... 0 A.. Squares... 0 A.. Rows... A.. Columns... A. Flag... A.. Squares... A.. Rows... 4 A.. Columns... 5 Appendix B Configurations of Marshal, general and flag... 6 Appendix C Counterpart configurations of marshal, general and flag... 5 C. Marshal and general in the same lane area... 5 C. Marshal and general in different lane areas... 5 C. Configurations without a counterpart Appendix D Configuration versus configuration of marshal, general and flag... 6 D. Marshal and general in the same lane area... 6 D. Marshal and general in different lane areas APPENDIX E Configurations of marshal, general, flag and colonel... 8 APPENDIX F Configurations of marshal, general, flag and spy APPENDIX G Configurations of flag and bomb G. Normal configurations of flag and bomb G. Special configurations of flag and bomb with a score > or = G.. Flag surrounded by bombs connected to bombs surrounding a low rank G.. Flag and a low rank surrounded by 4 bombs... 0 G. Special configurations of flag and bomb with a score <... G.. Flag and a low rank surrounded by bombs in a corner... G.. Flag surrounded by bombs in a corner with a mirror bomb structure... G.. Flag surrounded by bombs with a low rank surrounded by bombs... G..4 Double bomb defence... 4 Author: Han Wolf

6 G..5 Five bomb defence... Appendix H Configurations of flag, bomb, marshal and general... 4 H. Most used configurations... 4 H. Best performing configurations... 9 H.. Overview... 9 H.. Illustrations of score >=... 4 APPENDIX I Teams I. Marshal teams I.. Best performing middle teams I.. Most used middle teams I.. Best performing flank teams I..4 Most used flank teams I..5 Bombs in middle team I..6 Bombs in flank teams I..7 Marshal in or behind middle team I..8 Marshal in or behind flank team I. General teams... 5 I.. Best performing middle teams... 5 I.. Most used middle teams... 5 I.. Best performing flank teams... 5 I..4 Most used flank teams I..5 Bombs in middle team I..6 Bombs in flank teams I..7 General in or behind middle team I..8 General in or behind flank teams I. Marshal and general in the same domain I.. Best performing middle teams I.. Most used middle teams I.. Best performing flank teams I..4 Most used flank teams I..5 Bombs in middle teams I..5 Bombs in flank teams I..6 Marshal in or behind middle teams I..7 Marshal in or behind flank teams I..8 General in or behind middle team I..9 General in or behind flank teams Author: Han Wolf

7 6 Author: Han Wolf

8 Introduction. Problem definition In Stratego a player is completely free in the choice of a setup. Therefore the most simple way to get a setup in a Stratego program is to generate a setup at random. But total freedom of choice does not mean that all setups are equally good. Expert and also non-expert Stratego players have mentioned advices that distinguish good setups from bad ones. If such rules exist then a program to generate setups should comply to rules that are decisive for the quality of setups. The rules currently mentioned by experts may offer useable guidelines for human players but they do not suffice for the creation of a playable setup by a program. Only a few references contain specifications on a sufficient level of detail but these are fragmentary and far from complete. To make playable setups additional specifications are necessary. Besides a lack of detail and completeness there also is the more principal question of validity: give all mentioned advices really a good result?. The approach In order to find a base for the automatic creation of initial setups a substantial amount of work has been done in this area: Verification of advices found in literature A quest for criteria that are suitable for setup generation. The structure of this document reflects this approach. At first a survey is given of current literature and a statistical analysis of the validity of the advices. Only a minor part of profitable advices are suitable for implementation in a program..4 The Gravon database The Gravon database is a constantly growing source of information. The author has downloaded a snapshot of the years This snapshot contains only games between human players and has been used in the current analysis. No attempts have been done to gather snapshots of later years because they may be contaminated with games between human and computer players. Specifications of the Gravon database snapshot used for research Number of games 89 Number of moves Number of setups 4945 (from 5784 non-unique setups) Period ½ year (00-006) The volume of this snapshot may seem impressive. But for a really detailed and trustworthy analysis much more volume should be available. Therefore only a minor part of the results will be presented as really solid and trustworthy. From a theoretical point of view most results should be used with considerable discretion. The author has chosen the limit of 50 games as an arbitrary criterion for reliability of results. No attempt has been done to verify this limit by a mathematical analysis of the statistical data. 7 Author: Han Wolf

9 Advices for setups and their validity. Literature and websites Initial setups are discussed in literature and at some websites: Reference Where to find BG Advices from boardgames.com DC Winning Stratego EC Ed s Stratego Site (this site does not exist anymore) FM Forums on Yahoo.com and Boardgamegeek.com ST Stratego tips US Ultra Stratego VB Vincent de Boer: Invincible, A Stratego bot, 007: WP Wikipedia Stratego Vincent de Boer is an expert player, other advices may come from both experts and mediocre players.. Classification of advices A classification of the advices suggests that on a global level the following criteria affect the quality of a setup: Distribution of marshal, general and flag Protection of the flag Blockages Placement of scouts Placement of miners Cooperative units Miscellaneous other criteria. Most advices can be assigned to one of the criteria in this list. 8 Author: Han Wolf

10 . The distribution of marshal, general, spy and flag Reference Statements Confirmation / rejection DC Use marshal and general on different flanks for 40 % and Yes place colonels in the middle. DC Place marshal and general in the middle for 60 %. Only yes if the flag has been placed in the middle DC Place the spy in the vicinity of the general. More often no than yes EC ST US EC Place the marshal and general on different flanks. As often yes as no ST EC Choose a passive setup: More often no than yes Protect one of the flanks by general, spy, a colonel and bomb Protect the middle lane by marshal and a major Protect the other flank by a colonel, major and bombs. EC EC US The influence of majors and bombs has been disregarded. Place marshal, general and flag in the middle of the board and place misleading bomb structures on the flanks. The influence of bombs has been disregarded. Place marshal and general in the middle of the board because from there they can attain the flanks most efficiently if necessary. Yes Only yes if the flag has been placed in the middle EC Place one colonel on the third or fourth row for later use. Yes EC Protect a colonel on the first or second row by the marshal As often yes as no or the spy. EC Place the spy on the third row behind the general. More often no than yes US EC Place the spy in a lake column (prevents capture by a More often yes than no US scout). EC Place marshal and spy in the middle of the board. No US FM Place the spy in the middle More often no than yes FM Place the general in the middle, the colonels on the flanks and place the marshal on one of the back rows in order to protect miners on the back row More often yes than no VB Favourite places for high ranks are: Marshal: more yes than no Squares in the lake columns of the first or second row General: as often yes as no Squares in the lane columns of the second or third row Colonel: as often yes as no VB Do not place both colonels in the lake columns of first or More often no than yes second row. WP Place high ranks far from the flag in order to mislead the More often no than yes opponent. WP Place the spy near a general. More often no than yes WP Place the spy near a colonel. As often yes as no Probably the influence of vicinity of the spy depends on the strategy: the more risk a player accepts in moves of the general or colonel, the more vicinity of the spy is required. The influence of strategy during the moves of a game cannot be determined by a study of initial setups. 9 Author: Han Wolf

11 .4 The protection of the flag Reference Statement Confirm ALL Surround the flag by bombs As often yes as no BG BG EC FM Alternate between the corners, the rest of the back row and sometimes the middle rows or the lanes columns of the first row Use a double bomb defence 4 Some partial surroundings are good alternatives Front row: no Middle rows: no Corner: no Rest of back row: yes No or its mirrored counterpart EC Place a high rank near a flag surrounded by bombs. No answer possible Formulation is too vague EC Enclose a low rank in the bomb defence of the flag in a corner of the board No 4 or its mirrored counterpart 0 Author: Han Wolf

12 Reference Statement Confirm FM Include middle ranks in a bomb structure that surrounds Yes if not in a corner as in US the flag completely the previous advice. An example: 4 and similar kinds of setups FM Place the flag in the bottom middle Yes FM ST Place the flag on the back row Corner: no Rest of back row: yes US Do not use a double bomb defence Yes US Do not scatter bombs randomly over the board Yes US Surround the flag in a corner by bombs No 4 US or its mirrored counterpart Surround the flag at a side of the board by bombs No 4 or its mirrored counterpart Author: Han Wolf

13 Reference Statement Confirm US Do not use the flag in the front row surrounded by bombs Yes 4 US and similar kinds of setups Do not use the flag in the front row protected by bombs Yes 4 VB and similar kinds of setups Favourite squares for the flag are in the lake columns of the back row. VB Do not place the flag on a front row. Yes VB Protect the flag by sufficiently high and middle ranks, also if it is surrounded by bombs. As often yes as no Some other squares on the back row are good alternatives No answer possible Formulation is too vague Appendix F shows flag bomb configurations with their scores. Appendix G contains a discussion about protection of the flag by mobile ranks Author: Han Wolf

14 .5 Blockages, mobility and protection Reference Statements Confirmation BG EC FM US Mislead the opponent by units of bombs that surround a low rank instead of the flag. A variation is to place a mirror unit with the flag in the other corner of the board. No 4 4 BG ST US BG US Do not seal off a lane by bombs Yes (= do not) Place sometimes bombs on the row Yes for bomb in row No for other numbers DC Do not place pieces with the same rank adjacent to each other. Yes DC Do not place bomb structures in the middle area where they block moves from one side of the board to the other. Assume that only bombs in the rectangle C4 to H block moves from one side of the board to the other. 4 More often yes than no Reference Statements Confirmation EC Seal off two side lanes by bombs. No EC Place majors in the back rows for later use. No EC ST VB Protect bombs by a sergeant or a lieutenant. Yes for protection of, 4 or 6 bombs in a setup No for other numbers Author: Han Wolf

15 EC Do not block pieces by bombs or the flag. VB Determine the number of blocked paths per setup. Then the less the number of blocked paths is the less the score should be. High scores for 0, or 5 blocked paths per setup. Low scores for other numbers of blocked paths per setup FM Place the spy near the flag or in a corner of the board Only yes for the flag in the rear and middle of the board FM FM WP ST ST VB Sprinkle the bombs that do not surround the flag over the back rows; sometimes place one bomb in the front rows Protect pieces by at least one piece with a rank that is exactly two levels higher. Keep the front row mobile, therefore: only place low ranks in the front row do not place bombs in the front row. This has been analysed by deriving the scores for all setups with only lieutenants, sergeants, miners and scouts in the front row and with the restriction that miners do not occupy squares that give access to a lane. Do not block pieces in the back rows by other pieces with a one or two lower (= a less strong) rank. More often no than yes Yes No Yes for - or 5 or 6 not blocked pieces No for other numbers 4 Author: Han Wolf

16 Reference Statements Confirmation US Place misleading units of a low rank surrounded by bombs in a corner No and place the flag in the middle of the board and protect the flag by high ranks. US VB Do not surround a piece other than the flag by bombs. Yes But sergeant and miner are VB VB VB Do not place too many pieces of one rank on the front rows. This formulation is rather vague. Reformulate: one piece in the front row is too much for pieces with initially one piece; for other ranks more than one piece is too much. Place some lower ranks (not miners) in the lane columns of the front row. Enable the use of middle pieces in the back rows in descending order of rank (strong ranks before weak ranks). exceptions More often no than yes Yes for lieutenant or major/ - captains or sergeants / - scouts No for other ranks or other numbers Yes for -4 not blocked pieces No for other numbers VB Do not block pieces in the lane columns by the spy. More often yes than no VB Do not place the spy adjacent to a bomb. No Only yes for or adjacent bombs 5 Author: Han Wolf

17 .6 The placement of scouts Reference Statements Confirmation DC Save three to six scouts in the lane columns of the back More often no than yes row. High scores exist only for DC EC EC* US FM FM WP Save two or more scouts in the lane columns of the back rows. Place a few scouts on the back row in order to preserve them for the end game. Place scouts in the lane columns. Scatter the scouts over the board Interpretation: scatter scouts evenly over the board Place reserve scouts in the back rows. -4 scouts More often no than yes High scores exist only for - scouts No Only high scores for 0 scouts and scout Yes with much nuance No Scouts should not be distributed evenly over rows and columns Yes High scores exist for -4 scouts ST Use series of consecutive scouts in a row If in row and with a length of less than 7 : Yes Else: No VB Do not place a scout adjacent to a bomb. Yes s decrease as bombscout numbers increase *Additional remark Rather surprising are high scores for lake columns in the third row. Strategically scouts on these positions are meant for misleading actions. So the approach of misleading by both placement and moves may be a profitable one. 4 6 Author: Han Wolf

18 .7 The placement of miners Reference Statements Confirmation VB Do not place miners in the lane columns of the front row. More often yes than no Remarkable are the high scores for miners on A4 and K4 EC Place most miners in the two back rows. Yes High scores for 4-5 miners DC Save at least two miners in the back row and separate Yes them by at least one square. WP Place miners both for direct use in the front rows and for later use in the back rows. Yes for most balanced distributions FM Place miners on the back row Yes for - miners No for other numbers FM Keep at least of the 5 miners on the back rows Yes (4, 5) and no () High scores for 4-5 miners 7 Author: Han Wolf

19 .8 Cooperative units Reference Statements Confirmation EC Use units of 4 pieces consisting of: Only the front parts of US A high rank domains with a top rank Three lower ranks. have been analysed. An example of a cooperative unit: Colonel For data with 50 or more Scout games the answer is: Miner Sergeant. More often yes than no. Another example of a cooperative unit: Marshal Miner Scouts behind them. EC Do place ranks on positions where they can support a plan for action The setup area has been split up into 4 areas. 4 LEFT LANE AREA MIDDLE LANE AREA RIGHT LANE AREA BACK AREA The content of the front areas has been analysed by counting the number or ranks in these areas. The flank areas contain 6 pieces and the middle front area contains 8 pieces. The contents of a front area may constitute a team of cooperating pieces where low ranked pieces support one or two high ranked pieces. About 0 combinations of ranks occur more than 50 times in a game and constitute a team that consists of one or two high ranks together with low ranks. About 65% of teams consisting of a high rank with middle and low ranks have a score >. 8 Author: Han Wolf

20 .9 Advices without an analysis The following advices have not been analysed. Reference Statements Why no analysis? VB A setup may be created for and require a specific playing style. These advices refer EC Place the spy next to the general and keep an additional middle rank in the vicinity in order to protect the spy against scout to both setup and game moves. attacks. 9 Author: Han Wolf

21 .0 Conclusion: advices do not constitute a base for a Stratego program A quest for advices has produced a list of 75 advices. Most of them have been analysed. About 0 advices show a simple and consistent yes or no. The other advices show partly yes, partly no. Often the conditions for confirmation or negation are complex and contradictory. Only for a few simple results plausible explanations are available. Although 75 advices have been studied, there is a good chance that not all relevant features for the construction of good performing setups have been mentioned in literature and on websites. So the analysis has produced more questions than answers and the question arises was it worth all the effort spent to it? Indeed all this may seem unsatisfactory. But this statistical analysis of Stratego has shown the limitations of what experts say about does and don ts. A few simple rules have been detected that may guarantee better chances to win by features of a setup. To know that complex mixtures of yes and no answers exist may prevent disillusions. Anybody who knows this will not waste time and effort into the creation of low performing setups based on oversimplification and bad assumptions.. What to do when advices are not sufficient? How to use the results in a Stratego program? A number of about 0 confirmed advices with simple rules is too small. The content of the advices does not cover enough aspects of setups. In theory it is possible to extend this base with the more complex results from the analysis. Complex rules may be implemented by the use of decision tables. Then about fifty decision tables are required as a base for the generating of setups. Their relative weight has to be determined. How this should be done is uncertain. There is a good chance that a high degree of arbitrariness will exist in the assignment of relative weights. This does not look like an attractive solution for the generating of setups. And then there is the question of completeness. Probably not all relevant criteria are present in the collection of advices. This Gordian knot has to be chopped somehow. The author has chosen for a further research by additional statistical analysis of setups. In this analysis additional criteria have been studied. The choice for these criteria is based on personal interest and curiosity, more than on solid theoretical view. In the next chapter some basic aspects of setups are discussed. The chapters thereafter contain a discussion of the additional statistical analysis. The last chapter discusses what to do with the final results. 0 Author: Han Wolf

22 Basic concepts. Lanes The lanes offer access to the territory of the opponent. Efficiency requires that pieces should be moved preferably through their nearest lane. Author: Han Wolf

23 . Distances and domains This shows the number of moves that a piece on one of the 40 squares requires to attain the nearest lane. Any square has a nearest lane. A domain contains squares with the same nearest lane. Author: Han Wolf

24 These three domains and the distances of squares to the nearest lane are important properties of the board. They determine the potential of pieces to attain the territory of the opponent. They reflect the vulnerability of pieces to attacks by the opponent. Any theory of setups has to be based on these areas and distances.. Front and rear Pieces on the front rows (rows and 4) are involved with the first actions of a game. Their ranks and positions should be tuned to immediate action, both offensive and defensive. Essential is their resistance against early gambling attacks of the opponent. Pieces on the back rows are meant for later use and for direct defence of the flag. In most games the flag should be placed on a square of the back row (row ). Four strategies and corresponding areas should be considered in the creation of a setup: Separate local strategies for the front parts of the three domains One common strategy for the rear in the end game... Three separate initial strategies in the front parts of the domains In the initial phase of a game the division of the board into the front parts of the left, middle and right domain is highly relevant. During a long number of moves traffic of pieces between these areas is impossible because the paths between the areas are blocked by pieces. Therefore in the initial phase of the game separate local strategies can be applied to these three areas... One common final strategy When about half of the pieces have been removed from the board then the division of the board into the three areas loses relevance. As the number of pieces on the board diminishes the division into three areas finally plays no role at all: the defence of the flag and therefore paths to the flag grow far more relevant than any division into lane related areas. Author: Han Wolf

25 .4 Simple rank statistics of squares, rows and columns In literature some attention has been spent to simple statistics of ranks. Frequencies for ranks on squares have been mentioned because of their assumed applicability in the computation of unknown rank detection methods. In this document attention an extension is being given to these statistics by giving attention to the results of ranks on positions on the board. Not only squares but also row and column statistics have been computed from games in the Gravon database. These statistics can be found in Appendix A Basic rank statistics. The applicability of such statistics in a Stratego program is very restricted..5 Row statistics of the flag One of the row statistics appears to be a relevant one. Flag Row 4 0,49 0 0,7 05 0,86 465, This shows the validity of an overall rule: place the flag on the last row. 4 Author: Han Wolf

26 .6 Relevant criteria and stepwise refinement In the following chapters attention will be given to the domains. Their division into front parts and a rear part appears to be a (much) more relevant criterion for rank statistics. The front and rear domains will be combined with rank criteria: Top ranks: marshal, general and flag Sub top ranks: colonels and spy. The sequence will be: from a general level to a deeper level of detail. Additional attention will be given to the protection of the flag by bombs. The deepening of the level of detail goes stepwise. Each level of detail gives a group setups that wins and a group that loses. Each additional criterion refines the view on the capacity to win or lose by splitting up the group of winning setups into both winning and losing subgroups and additionally splitting up the group of losing setups into both winning and losing subgroups. More aggregated level of detail An additional criterion Wins Loses The refinement is especially effective if the green part (score >=) increases by the addition of a criterion, because this enlarges the range of favourite setups. If the red part increases than this too will improve the choice of good setups. But the unpredictability of setups requires as broad a range as possible, therefore an increase of the green part is a more favourable one. 5 Author: Han Wolf

27 4 The distribution of top ranks over the board After the musing about basic setup concepts this is the right place to show additional characteristics of setups used by human players. Here follow some studies concerning the relative positions of marshal, general and flag. 4. Marshal and general together or apart Marshal and General in the same domain Win Loss Draw Yes ,88 No ,06 This statistical overview shows that setups with marshal and general in the same domain are losing ones. This level of detail suggests that a Stratego player should not use setups with marshal and general in the same domain. 4. Marshal and general front or back Marshal Front General Front Win Loss Draw N Y ,6 Y Y Y N ,96 N N ,79 This shows the validity of two rules: Place both marshal and general in the front domains Or place the general in the front domains and the marshal in the rear. 6 Author: Han Wolf

28 4. Marshal, general and flag Marshal and General in the Marshal and Flag General and Flag same area in the same area in the same area Win Loss Draw Yes Yes ,96 No ,77 No Yes No ,4 No Yes ,06 No No ,89 This shows the validity of two rules: Do not place marshal and general in the same domain Place the flag in a domain that contains a top rank (marshal or general). This overview shows that the rule in the previous paragraph is an oversimplification. The position of the flag has an influence on the score. Not all setups with marshal and general in separate domains will have more wins than losses. 4.4 Marshal, general, flag and front versus rear The global position of the top ranks in rows can be taken into account. This gives additional criteria. Marshal and General in the Marshal and Flag General and Flag Marshal in front General in front same area in the same area in the same area rows rows Win Loss Draw Yes Yes Yes Yes ,00 Yes No ,85 No Yes ,86 No No ,66 No Yes Yes ,75 Yes No ,9 No Yes ,87 No No ,68 No Yes No Yes Yes ,4 Yes No ,5 No Yes ,5 No No ,8 No Yes Yes Yes ,06 Yes No ,78 No Yes ,40 No No ,77 No No Yes Yes ,85 Yes No ,98 No Yes ,99 No No ,9 Not very surprisingly this further subdivision of setups leads to more cases with a chance of success. Even in setups with marshal and general in the same domain a kind of setup emerges as a profitable one, namely setups with marshal, general and flag in the same domain and both marshal and general in the front rows. 7 Author: Han Wolf

29 4.5 Detailed views of marshal, general and flag The division of the board into domains and front and rear facilitates statistics with additional criteria: domain where the marshal resides (left, middle, right and front or rear) domain where the general resides (left, middle, right and front or rear) domain where the flag resides (protected by no top rank, top rank or top ranks). These additional criteria lead to a total of 96 separate cases. On this level of detail not all configurations of marshal, general and flag are based on data volumes large enough to give trustworthy results. But most (89) cases are based on 50 or more games and that has been reason enough to make a study of this level of detail. The next paragraphs contain descriptions of configurations based on 50 or more games. With this level of detail the limits of the Gravon database have been attained. Even trustworthy results based on 50 or more games should be handled with considerable restraint. Most results in more detailed studies are based on smaller numbers of games. The lack of sufficient volume makes more detailed results very untrustworthy. 8 Author: Han Wolf

30 4.5. Symmetrical configurations Here are 4 most frequently used configurations: , , , ,95 Remark: flags and areas have been shown where both parts of the rear may contain the flag. Symmetrical configurations have both top ranks in the middle of the board. The flag can be protected by both ranks or it is not protected at all. This concentration of top ranks is used in less games than the dispersion of top ranks over different areas. Symmetrical configurations have more losses than wins. The only exception (with a slight advantage) is the configuration with marshal and general in the middle front area and the flag behind these top ranks. An overview of all possible configurations can be found in appendix C. 9 Author: Han Wolf

31 4.5. Mirror configurations with marshal and general in the same domain If marshal and general reside in the same flank domain then each configuration has a mirrored counterpart. Here are 4 most frequently used configurations with their mirrored counterparts: 4 64, , , , ,94 4 7, , ,88 Remark: flags and areas have been shown where both parts of the rear may contain the flag. Concentrations of top ranks on the flanks constitute a not much used minority. A first impression is that the results do not follow simple comprehensible rules. Some mirror cases behave like expected by showing similar scores. But why have some mirror configurations contrary scores? The overview of results in this paragraph gives more questions than answers. 0 Author: Han Wolf

32 Frequency is not the only interesting aspect of configurations. A focus on the scores may give additional insight. Here are the four configurations with the highest scores and their mirrored counterparts. 4 64, ,78 4 0,04 4, ,94 4 7, , ,88 Remark: flags and areas have been shown where both parts of the rear may contain the flag. But here too mirrored counterparts can have contrary scores. This learns a lesson: it is not a good idea to implement random choice of mirrored counterparts, if one of the mirrored setups has a score greater than or equal to and the other one may have a score less than. Author: Han Wolf

33 4.5. Mirror configurations with marshal and general in different domains If marshal and general reside in the different domains then each configuration has a mirrored counterpart. Here are 4 most frequently used configurations with their mirrored counterparts: 4 04, 4 65,8 4 0, , 4 607,8 4 90, , ,94 Here more consistency between configurations and their mirrored counterparts is visible. Next page shows the four configurations with the highest scores and their mirrored counterparts. Author: Han Wolf

34 4 550, ,7 4 4, , ,5 4 77, , ,7 The best performing configurations with distributed top ranks have a general in the front rows and a marshal in the back rows. An explanation may be that it is important keep the threat of an unknown marshal intact as long as possible. The flag is placed together with one top rank in the left, middle or right side of the board. Appendix C contains an overview of all possibilities. The complete overview contains other inconsistent cases. But although the inconsistent cases constitute a minority, there too is a disturbing lack of good explanations. Author: Han Wolf

35 4.5.4 Setup against setup What setup will win against any other setup? Probably an answer to this question can be found by a study of configuration versus configurations statistics. Unfortunately most results of configuration against configuration miss sufficient volume. But for a definite part of the configurations enough volume exists. Here follow some examples of setups played by red against blue with more or less trustworthy results. 4 6, 4 Win 09 Loss 48 Draw ,0 Win 45 Loss 60 Draw ,7 85,56 4 Win 946 Loss 79 Draw ,7 7,5 4 Win 644 Loss 550 Draw 48 4,6 85, 4 Win 78 Loss 769 Draw ,94 The first example shows a concentration of top ranks in the middle area. The overall score of,0 makes this configuration not a great winner. 4 Author: Han Wolf

36 Here an example again, this time a configuration with a better overall score. 4 49,75 4 Wins 9 Losses 608 Draws 0 65,8 Wins 946 Losses 79 Draws ,7 4,40 4 Wins 09 Losses 48 Draws ,0 77,6 4 Wins 6 Losses 057 Draws 97 90,6 97, 4 Wins 587 Losses 64 Draws 8 0,5 This overview shows both wins and losses against other configurations. A study of all configurations shows that most configurations have both wins and losses against other configurations. Some configurations lose against any other configurations. An overview of all configurations can be found in Appendix D. 5 Author: Han Wolf

37 4.6 Summary and conclusion A study of the relative positions of marshal, general and flag has not given simple rules for the making of setups. But some overall tendencies can be detected: Do not place marshal and general together in the left, middle or right domain of the board Use configurations with general in the front rows and marshal in the back rows of the board Second best are configurations with both general and marshal in the front rows of the board Place flag and at least one top rank together in the left, middle or right domain of the board The chance of success of a configuration is about the same as its mirrored counterpart. But these simple rules have known exceptions. Another result of this study is the simple rule that no configuration is invulnerable. Every configuration can be beaten by at least one other configuration. If a program implements only one configuration with the highest score this would make a program too predictable. In the next chapter the limitations of the Gravon database will be felt still more than in the current chapter. Many of the cases in the current chapter have a sound base of 50 or more games. That will not be the case in the next chapter. 6 Author: Han Wolf

38 5 The distribution of sub-top ranks over the board Marshal, general and flag together constitute the top ranks on the board. Sub-toppers are colonel, spy and bomb. Strategically these ranks have a significant influence. But their impact is less than the influence that comes from the top ranks. The top-ranks ranks with the sub-toppers together constitute the backbone of a setup. 5. The influence of colonels Should colonels be placed in the same area as the marshal or general? An answer may be found by the addition of the number of colonels to distributions of marshal, general and flag. 5.. Rules for marshal and general in the same domain Marshal & Flag Nr Colonel Win Loss Draw Same domain , , ,88 Different domain , , ,68 The deeper analysis in this paragraph leads to further refinement. The rule Do not place marshal and general in the same domain (mentioned in Pargr 4.) appears to have an exception for Marshal, flag and one colonel in the same domain. 7 Author: Han Wolf

39 5.. Rules for the marshal and general in different domains Marshal & Flag General & Flag Nr Colonel Marshal Nr Colonel General Win Loss Draw Same domain ,00 74,00 0, ,00 5,00 0, ,00 966,00, , ,00, ,00 64,00, ,00 59,00, Different domain Same domain , , , , , , Different domain , , , , , ,95 Other rules are in force if marshal and general are spread over different domains. If the flag occupies the same domain as the marshal Two colonels may be placed in the same domain as the general Two colonels may be placed in the same domain as the marshal One colonel may be placed in both the marshal and the general domain One colonel may be placed in the same domain as the marshal Although there exists more than one option apparently the best choice is to place both colonels in the same domain as the marshal. If the flag occupies the same domain as the general Two colonels may be placed in the same domain as the marshal One colonel may be placed in both the marshal and the general domain One colonel may be paced in the same domain as the marshal Here too more than one option exists and here too the best choice is to place both colonels in the same domain as the marshal. If the flag is in the domain without marshal and general The two colonels should occupy the same area as the flag. These rules have one common aspect: the defence of the flag requires at least two colonels or higher, but often even three high ranks are required. 8 Author: Han Wolf

40 5.. Significant cases on a deeper level of detail The more detailed analysis of distributions of marshal, general, flag and colonels has given 97 results based on less than 50 games and 48 results based on 50 or more games. Here 4 configurations with the highest frequencies are shown together with their mirrored counterparts ,55 4 0,9 4 60, ,8 4 55, 4 9,5 4 5,4 4 59, A complete overview the 48 results can be found in Appendix E. Only a fraction ( percent) of the results is relevant enough for use in a program. Frequencies of the counterparts show major differences. Why do human players tend to use only one of the counterparts? If the reason is psychological then this phenomenon lies outside the scope of this study. 9 Author: Han Wolf

41 5. The influence of the spy Should the spy be placed next to the general? This property has been added to already studied configurations of marshal, general and flag. 5.. Configurations ordered by frequency The 4 most occurring configurations are shown here. Spy next to general Otherwise 4 Win Loss Draw Win Loss Draw , ,00 Spy next to general Otherwise 4 Win Loss Draw Win Loss Draw , ,69 Spy next to general Otherwise 4 Win Loss Draw Win Loss Draw , , Spy next to general Otherwise 4 Win Loss Draw Win Loss Draw , ,9 The results do not provide a simple rule for the presence of the spy near the general. 40 Author: Han Wolf

42 5.4. Configurations ordered by score The 4 configurations with the highest scores are shown here. Spy next to general Otherwise 4 Win Loss Draw Win Loss Draw , ,95 Spy next to general Otherwise 4 Win Loss Draw Win Loss Draw , ,6 Spy next to general Otherwise 4 Win Loss Draw Win Loss Draw , ,4 Spy next to general Otherwise 4 Win Loss Draw Win Loss Draw , ,06 These high performing configurations show a strong preference for the placing of the spy near the general. The complete overview of configurations in sequence of score can be found in appendix F. The sum of all setup results in appendix F: Sum of shown setup configurations Spy next to general Otherwise Win Loss Draw Win Loss Draw , ,0 The total overview shows that the positive correlation shown in this paragraph is absent in many other configurations Additional remarks The placement of a spy next to the general is necessary if the program uses an offensive strategy wherein the general may be exposed to an attack by the marshal of the opponent. In most games human players use a wait-and-see strategy wherein the ranks of marshal and general are kept hidden as long as possible. Then the place of the spy is not governed by the simple rule place the spy next to the general. 4 Author: Han Wolf

43 6 Protection of the flag by bombs Areas on the board are defended actively by the top ranks. Bombs constitute the passive way of defence. As long as the places of bombs are unknown the threat to lose a piece on a bomb prevents effectively strategies that are based on long series of gambling attacks by the opponent. What are the best configurations of the flag and bombs? The following overview shows how the score of configurations depends on the position of the flag and the surrounding of the flag by bombs. Square Nr bombs Win Loss Draw Corner , , ,90 Edge , , , ,9 Middle , , , , ,58 This level of detail suggests that the flag should only be placed on the back row. Corner positions give a slight advantage if no bombs are placed next to the flag. Other positions of the back row only give an advantage if the number of bombs next to the flag equals or. It is possible to split up the categories. The following paragraphs give a more detailed look at flag bomb configurations. But only positions of the flag on the back row are considered. In theory positions in the left and right columns of the board may give a slight advantage, but such setups require completely specific strategies. For the making of a Stratego program it is better to focus on less specific setups. 4 Author: Han Wolf

44 6. Flag bomb configurations ordered by frequency The 4 most used configurations together with their mirrored counterparts are shown here. 0,95 0,86 4 Win Win 9 Loss 87 Loss 55 Draw 65 Draw ,9,8 4 Win Win 5 Loss 50 Loss 6 Draw Draw ,55,7 4 Win 47 4 Win 08 Loss 97 Loss 770 Draw 89 Draw ,95 0,7 4 Win Win 8 Loss 9 Loss 8 Draw 67 Draw The simple rule place the flag on an edge square of the back row surrounded by three bombs or next to one bomb gets a nuance. Also a study of all configurations shows considerable nuances. There are no simple explanations for deviations from the simple rule. 4 Author: Han Wolf

45 6. Flag bomb configurations ordered by score The 4 most best performing configurations based on 50 or more games together with their mirrored counterparts are shown here.,59,06 4 Win 0 4 Win 8 Loss 68 Loss 78 Draw Draw 0 8 7,55,7 4 Win 47 4 Win 08 Loss 97 Loss 770 Draw 89 Draw ,4 0,87 4 Win 9 4 Win 59 Loss 96 Loss 69 Draw 0 Draw ,,6 4 Win 50 4 Win 589 Loss 78 Loss 60 Draw 7 Draw Also here considerable nuance appears to be necessary for the more simple rules found on a higher level of aggregation. A complete overview of flag-bomb configurations can be found in Appendix G. 44 Author: Han Wolf

46 6. Flag bomb configurations combined with marshal general configurations Flag bomb configurations may be differentiated further by combining them with configurations of marshal and general. This leads to 4076 combinations of which 0 are based on 50 or more games. Appendix H contains the collection of cases based on 50 or more games. 6.. Most used marshal-general-flag-bomb configurations Left Middle Right ,76 Wins 487 Loss 654 Draw 44 Left Middle Right 064 4,6 Wins 556 Loss 474 Draw 4 Left Middle Right 05 4,46 Wins 599 Loss 400 Draw 6 Left Middle Right 0 4,0 Wins 496 Loss 48 Draw 4 Left Middle Right 97 4, Wins 467 Loss 4 Draw 8 The next page shows another 5 cases. 45 Author: Han Wolf

47 Left Middle Right 747 4,68 Wins 460 Loss 64 Draw Left Middle Right 78 4,7 Wins 40 Loss 0 Draw 7 Left Middle Right ,88 Wins 0 Loss 44 Draw 9 Left Middle Right ,6 Wins 8 Loss 400 Draw 6 Left Middle Right 65 4,06 Wins Loss 0 Draw 8 This paragraph let see in more detail what kind of setup most human players prefer to use. The combinations on this page confirm the paragraph 6.. Most setups contain the flag surrounded by bombs. 46 Author: Han Wolf

48 6.. Best performing marshal-general-flag-bomb configurations Left Middle Right Wins 78 Loss 9 Draw Left Middle Right 4, Wins 70 Loss Draw Left Middle Right 4, Wins 65 Loss 5 Draw 0 Left Middle Right, Wins 74 Loss Draw 0 Left Middle Right, Wins 6 Loss 7 Draw This paragraph comes with a surprise. It shows unexpectedly the good performance of a flag bomb configuration at the side of the board. Thereafter come more normal constructed setups. The next page shows another 5 cases. 47 Author: Han Wolf

49 Left Middle Right, Wins 4 Loss Draw Left Middle Right, Wins 00 Loss 8 Draw Left Middle Right, Wins 6 Loss Draw Left Middle Right, Wins 4 Loss 5 Draw Left Middle Right, Wins 9 Loss 5 Draw Why these setups perform well is an open question. Appendix H shows the complete collection. It may give inspiration to authors who are in search for a collection of setups with good perspectives and enough diversity for the sake of unpredictability. But generally applicable explanations for good or bad behaviour are totally missing. 48 Author: Han Wolf