Winning strategy - Revision history

Selinger: Removed broken link, removed stub warning

2021-09-12T01:06:07Z

Removed broken link, removed stub warning

Selinger: Added a category

2020-12-28T22:00:50Z

Added a category

Selinger: Added a definition, expanded examples, and added some proof sketches.

2020-06-14T03:03:20Z

Added a definition, expanded examples, and added some proof sketches.

Halladba: added a heading and sections to develop

2008-02-07T00:28:47Z

added a heading and sections to develop

Halladba: added cat. theory

2008-01-31T15:09:42Z

added cat. theory

Roland Illig: + some thoughts

2007-06-13T14:28:49Z

+ some thoughts

New page

In [[Hex]], there exists a winning strategy. When you play without the [[swap rule]], the first player can always win. When you play with the swap rule, the second player can always win.

The reason for the second player's sure win strategy is simple: After the first player's move, there are two possibilities:
# The move was so good that no matter what the second player does, the first will win. In this case, the second player should swap colors.
# The move was bad, so that the first player cannot win. Because the game cannot end in a draw, this means that the second player can win. In this case, don't swap colors.

The only remaining problem is to decide whether the situation after the first move can be won by the first player or not.

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 On an asymmetric board of size ''n''×''m'', where ''n'' ≠ ''m'', the player with the shorter distance between their edges always has a winning strategy, no matter who goes first. In this case, a [[Theory#Winning_strategy_for_non-square_boards|particular winning strategy]] is known.
-There are some positions on non-empty boards where it is known which player has a winning strategy. Two such positions are the [[a1 opening]] and the [[b1 opening]], both of which are winning for Blue. Also, any [[Theory#No_winning_strategy_for_Blue|symmetric position]], i.e., any position that looks the same from Red's and Blue's point of view (with the sides and colors flipped), is a win for the next player to move. Finally, if some position is winning for a player, then adding additional pieces of that player's color to the board, or removing pieces of the opponent's color, yields a position that is still winning for the same player. For this reason, we say that extra pieces of a player's own color "can only help" the player.
+There are some positions on non-empty boards where it is known which player has a winning strategy. Two such positions are the [[a1 opening]] and the [[b1 opening]], both of which are losing. Also, any [[Theory#No_winning_strategy_for_Blue|symmetric position]], i.e., any position that looks the same from Red's and Blue's point of view (with the sides and colors flipped), is a win for the next player to move. Finally, if some position is winning for a player, then adding additional pieces of that player's color to the board, or removing pieces of the opponent's color, yields a position that is still winning for the same player. For this reason, we say that extra pieces of a player's own color "can only help" the player.
-−
-==See Also==
-−
-[[strategy-stealing argument]]
 [[category: Theory]]
 [[category: Definition]]

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 [[strategy-stealing argument]]
-[[category:Theory]]
+[[category: Theory]]
 {{stub}}

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 ==Definition==
-==Existence of a winning strategy in Hex==
+A ''strategy'' for a player is a rule that specifies the player's next move from every possible position. If we fix a strategy for Red and a strategy for Blue, then the winner of the game is determined, because we can simply play the game to the end, using each player's strategy.
-In [[Hex]], there exists a winning strategy. When you play without the [[swap rule]], the first player can always win. When you play with the swap rule, the second player can always win.
-The reason for the second player's sure win strategy is simple: After the first player's move, there are two possibilities:
+A strategy for Red is called a ''winning strategy'' if it wins against every possible strategy for Blue. Similarly, a strategy for Blue is called a winning strategy if it wins against every possible strategy for Red.
 # The move was so good that no matter what the second player does, the first will win. In this case, the second player should swap colors.
 # The move was bad, so that the first player cannot win. Because the game cannot end in a draw, this means that the second player can win. In this case, don't swap colors.
-The only remaining problem is to decide whether the situation after the first move can be won by the first player or not.
+To take advantage of this fact, the second player would have to know which opening moves are winning for Red and which ones are winning for Blue, in order to figure out whether to swap or not. The second player would then also have to know a particular winning strategy, so that she does not accidentally lose by making a bad move. Since no such particular strategy is known, each player has to do the best they can, which is why the game is interesting to play.
 ==See Also==

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 In [[Hex]], there exists a winning strategy. When you play without the [[swap rule]], the first player can always win. When you play with the swap rule, the second player can always win.
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 The only remaining problem is to decide whether the situation after the first move can be won by the first player or not.
 [[category:Theory]]

← Older revision		Revision as of 15:09, 31 January 2008
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	The only remaining problem is to decide whether the situation after the first move can be won by the first player or not.		The only remaining problem is to decide whether the situation after the first move can be won by the first player or not.
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		+	[[category:Theory]]