![]() In a paper posted online last September, they proved that no method of adapting strategies in response to previous games - no matter how commonsensical, creative or clever - will converge efficiently to even an approximate Nash equilibrium for every possible game. Now, Rubinstein and Yakov Babichenko, a mathematician at the Technion-Israel Institute of Technology in Haifa, have explained why. But for each such mechanism, he said, “there are simple games you can construct where it doesn’t work.” “Economists have proposed mechanisms for how you can converge to equilibrium,” said Aviad Rubinstein, who is finishing a doctorate in theoretical computer science at the University of California, Berkeley. Yet attempts to find such efficient learning methods have always come up dry. But if people can play repeatedly, perhaps they could learn from the early rounds and rapidly steer themselves toward an equilibrium. This is especially the case if - as is typical in the real world - each player knows only how much she herself values the game’s different outcomes, and not how much her fellow players do. If people play a game only once, it is often unreasonable to expect them to find an equilibrium. “They use these equilibrium concepts, and they’re analyzing them as if people will be at equilibrium, but there isn’t always a satisfying explanation of why people will be at Nash equilibrium as opposed to just groping around for one.” “It has always been a thorn in the side of microeconomists,” said Tim Roughgarden, a theoretical computer scientist at Stanford University. But how do players get to equilibrium in the first place? In contrast with, say, a ball rolling downhill and coming to rest in a valley, there is no obvious force guiding game players toward a Nash equilibrium. When the Clear method is called, the cursor automatically scrolls to the top-left corner of the window and the contents of the screen buffer are set to blanks using the current foreground background colors.When players are at equilibrium, no one has a reason to stray. Using the Clear method is equivalent invoking the MS-DOS cls command in the command prompt window. 'This example produces the following results: ' Draw the right side, from bottom to top. ' Draw the bottom side, from left to right. ' Draw the left side of a 5x5 rectangle, from top to bottom. ' Clear the screen, then save the top and left coordinates. Protected Shared Sub WriteAt(s As String, x As Integer, y As Integer) This example produces the following results: With :? ArgumentOutOfRangeException as e -> WriteAt("-", 3, 0) // shortcut: WriteAt("-", 1, 0)Ĭonsole.SetCursorPosition(origCol + x, origRow + y) Protected static void WriteAt(string s, int x, int y)Ĭonsole.SetCursorPosition(origCol+x, origRow+y) Draw the top side, from right to left. Draw the right side, from bottom to top. Draw the bottom side, from left to right. Draw the left side of a 5x5 rectangle, from top to bottom. Clear the screen, then save the top and left coordinates. GetKeyPress "Select Background Color (W for White, G for Green, M for Magenta): " If validChars |> List.exists (fun ch -> ch.Equals(Char.ToUpper keyPressed.Ke圜har)) then If (Array.Exists(validChars, ch => ch.Equals(Char.ToUpper(keyPressed.Ke圜har)))) Private static Char GetKeyPress(String msg, Char validChars) Restore the default settings and clear the screen. If (Char.ToUpper(GetKeyPress("Display another message (Y/N): ", new Char )) = 'N') String textToDisplay = Console.ReadLine() Save colors so they can be restored when use finishes input.ĬonsoleColor dftForeColor = Console.ForegroundColor ĬonsoleColor dftBackColor = Console.BackgroundColor ĬonsoleColor newForeColor = ConsoleColor.White ĬonsoleColor newBackColor = ConsoleColor.Black Ĭhar foreColorSelection = GetKeyPress("Select Text Color (B for Blue, R for Red, Y for Yellow): ",Ĭhar backColorSelection = GetKeyPress("Select Background Color (W for White, G for Green, M for Magenta): ",Ĭonsole.Write("Enter a message to display: ") If the user chooses not to exit the program, the console's original foreground and background colors are restored and the Clear method is called again before re-executing the loop. ![]() The following example uses the Clear method to clear the console before it executes a loop, prompts the user to select a foreground and background color and to enter a string to display.
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