Nash equilibrium - Revision history

Xseda07: /* Dominant strategies */

2013-01-25T17:30:44Z

‎Dominant strategies

Xseda07: /* Prisoner´s dilemma */

2013-01-25T17:29:09Z

‎Prisoner´s dilemma

Xseda07: /* Definition of Nash equilibrium */

2013-01-25T17:28:12Z

‎Definition of Nash equilibrium

Xseda07: /* Definition of Nash equilibrium */

2013-01-25T17:27:09Z

‎Definition of Nash equilibrium

Xseda07: /* Games in strategic form */

2013-01-25T17:24:35Z

‎Games in strategic form

Xseda07: /* Games in strategic form */

2013-01-25T17:24:21Z

‎Games in strategic form

Xseda07: /* Games in strategic form */

2013-01-25T17:23:14Z

‎Games in strategic form

Xseda07: /* Definition of Nash equilibrium */

2013-01-25T17:22:17Z

‎Definition of Nash equilibrium

Xseda07: /* Definition of Nash equilibrium */

2013-01-25T17:21:58Z

‎Definition of Nash equilibrium

Xseda07: /* Definition of Nash equilibrium */

2013-01-25T17:20:53Z

‎Definition of Nash equilibrium

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 ==Dominant strategies==
-If we look at the matrix above, we can see that player one will never choose action <math>a_1</math>(of course in case, he is rational, which is expected). This is because all payoffs in first row are smaller than payoffs in second row. Also player two won´t choose action  <math>b_3</math> because all payoffs in this column are smaller than in other two columns.(Payoffs of Player 2 are -(payoffs) of Player 1). So we can delete it with clear conscience.<ref name="DLOUHY"/> This situation is showed at the picture below.
+If we look at the matrix above, we can see that player one will never choose action <math>a_1</math>(of course in case, he is rational, which is expected). This is because all payoffs in first row are smaller than payoffs in second row. Also player two won´t choose action  <math>b_3</math> because all payoffs in this column are smaller than in other two columns.(Payoffs of Player 2 are -(payoffs) of Player 1). So we can delete it with clear conscience.<ref name="DLOUHY"/> This situation is showed in the picture below.
 [[File:Matrixdeleted.gif]]
 <div>

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 The [[Prisoner's dilemma]] is one of the most well-known strategic games. It was researched many times and many ways due to huge variety of situations in real life which is similar to this problem.
 <div>
-'''The definition of Prisoner´s dilemma'''
+'''The definition of Prisoner´s dilemma'''<br/>
 There are two suspects in a major crime situated in separate cells. We have enough evidence to convict each of them of a minor offense, but we can´t convict either of them of the major crime unless one of them pleads guilty and tells information that help to find the second one guilty. If they both say nothing (cooperate), each will be convicted of the minor offense and spend one year in jail. If one and only one of them betrays other, than he will be set free and used as a witness. The other one will spend four years in jail. If they both betray, they will spend three years in jail. <ref name = "NASH"/> <br/>
 [[File:Prisoners_dilemma.gif]]
 We can demonstrate this situation as shown above. Payoffs in the matrix represent years in prison (higher values are worse). When we try to find Nash equilibrium, we get pair of action '''(betray,betray)''', but as we can see that isn´t the  best solution. The Best solution is '''(cooperate, cooperate)'''. This example shows, that Nash equilibrium is not always Pareto-effective.
 ===Bach or Stravinsky (also known as battle of sexes)===
 Two people decided to go out together to a concert of music by either Bach or Stravinsky. However, one person prefers Stravinsky and the other person prefers Bach.                     <ref name="CANETTI"> [ROSEN, Alon a Ran CANETTI. Cryptography and Game Theory: Lecture 6: Basics of Game Theory. [online] http://www.cs.tau.ac.il/~canetti/f09-materials/f09-scribe6.pdf [cit. 2013-01-25]. ] </ref>

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 Now We can define Nash equilibrium as an action profile <math>a*</math>  with the property that no player <math>i</math> can do better by choosing an action different from <math>a*_i</math> , given that every other player  <math>j</math> adheres to <math>a^*_j</math> .<ref name="NASH"></ref>
 <div>
-or to be absolutely exact:<br/>
+Or to be absolutely exact:<br/>
 Let <math>a</math> be an action profile, in which the action of
 each player <math>i</math> is <math>a_i</math>. Let <math>a'_i</math> be any action of player <math>i</math> (either equal to <math>a_i</math>, or different from it). Then <math>(a'_i , a_-_i)</math> denotes the action profile in which every player <math>j</math> except

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 <math>i</math> chooses her action <math>a_j</math> as specified by <math>a</math>, whereas player <math>i</math> chooses <math>a'i</math>. (The <math>-i</math>
 subscript on a stands for except <math>i</math>.) That is, <math>(a'_i , a_-_i)</math> is the action profile in which
-all the players other than <math>i</math> adhere to a while <math>i</math> deviates to <math>a'_i</math>. (If <math>a'_i = a_i</math> then
+all the players other than <math>i</math> adhere to <math>a</math> while <math>i</math> deviates to <math>a'_i</math>. (If <math>a'_i = a_i</math> then
 of course <math>(a'_i , a_-_i) = (a_i, a_-_i) = a</math>).

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 * a set of '''players <math>i</math>'''
 * for each player, a set of '''actions''' (or strategies) '''<math>a_i</math>'''
-* for each player, '''preferences''' <math>u_i</math>(or payoffs) over the set of '''action profiles'''<math>a</math>
+* for each player, '''preferences'''  <math>u_i</math>(or payoffs) over the set of '''action profiles'''  <math>a</math>
 <div>
 Games in strategic form are very often represented by payoff (or we can say table) like this one below.<br/>

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 To define it more precisely, strategic game consist of:<ref name="NASH">[OSBORNE, Martin J. Draft chapter from An introduction to game theory [online] http://www.economics.utoronto.ca/osborne/igt/nash.pdf. Version: 2002/7/23. s. 11-52 [cit. 2013-01-25].]</ref>
 * a set of '''players <math>i</math>'''
-* for each player, a set of '''actions''' (or strategies) '''<math>a</math>'''
+* for each player, a set of '''actions''' (or strategies) '''<math>a_i</math>'''
 * for each player, '''preferences''' (or payoffs) over the set of action profiles
 <div>

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 subscript on a stands for except <math>i</math>.) That is, <math>(a'_i , a_-_i)</math> is the action profile in which
 all the players other than <math>i</math> adhere to a while <math>i</math> deviates to <math>a'_i</math>. (If <math>a'_i = a_i</math> then
-of course <math>(a'_i , a_-_i) = (a_i, a_-_i) = a.)</math>
+of course <math>(a'_i , a_-_i) = (a_i, a_-_i) = a</math>.)
 The action profile <math>a^*</math> in a strategic game with ordinal preferences is a '''Nash equilibrium'''