CONSOLIDATION,EXTENSION AND DISCUSSION

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CONSOLIDATION,EXTENSION AND DISCUSSION

• BY
• BORA SOYTOPRAK

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CONSOLIDATION,EXTENSION AND DISCUSSION

• Despite the considerable appeal of the Nash
  equilibrium concept numerous criticisms of the
  concept have come from both experimentalists and
  theorists. The debate has asked whether the Nash
  concept is too imprecise, whether it requires too
  much calculation, whether it can be based solely
  on rationality, whether it accounts properly for
  risks, and whether the expected payoff
  maximization assumption is reasonable.

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VALIDITY OF THE NASH EQUILIBRIUM CONCEPT

• There are some serious doubts remain unresolved
  about the Nash equilibrium concept indicating
  that Game Theory is not yet a settled science.
  Even this should give encouragement not the
  opposite, to future Game Theorists like us, for
  it shows that there is a lot of room for new
  thinking and new research in the subject. We
  should not forget that a totally settled science
  would be a dead science.

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VALIDITY OF THE NASH EQUILIBRIUM CONCEPT

• Most of the games are non-cooperative, in tha
  sense that every player takes his action
  independently.
• Nash equilibrium has just the property of the
  simultaneous best responses. In any purported
  that is not a NE at least one player could have
  done better by switching to a different action.

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VALIDITY OF THE NASH EQUILIBRIUM CONCEPT

• Now the criticism is when asked why players
  should act in a game as in some NE.
• The response to this question is why not?
• We illustrate this rebuttal using an example with
  provided figure on the next slide.

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VALIDITY OF THE NASH EQUILIBRIUM CONCEPT
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VALIDITY OF THE NASH EQUILIBRIUM CONCEPT

• The unique Nash equilibrium is (A , A) yielding
  the payoffs (2 , 2) .
• However playing C also guarantees us that same
  payoff as we would get in that NE.
• But it is also possible returning back to NE if
  we prolong the game.

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IS THE NE TOO IMPRECISE?

• The criticism is based on the observation that
  many games have multiple Nash equilibria,
  therefore the concept fails to pin down outcomes
  of games sufficiently precisely to give unique
  predictions.
• In some games one of the many Nash equilibria
  could emerge as a focal point if the players
  expectations could converge on it.

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IS THE NE TOO IMPRECISE?

• Focal points are often governed by these
  features that assist the players expectations to
  converge
• historical
• cultural
• linguistic

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DO PLAYERS IN ACTUAL GAMES PLAY NE STRATEGIES?

• The criticism simply says that NE is unrealistic
  as a description of the outcomes of actual games.
• The critics argue that the concept of a NE is too
  subtle and the calculation of NE strategy in an
  actual game to difficult for players in real life
  games.

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DO PLAYERS IN ACTUAL GAMES PLAY NE STRATEGIES?

• Researchers have also conducted numerous
  experiments to test how people act in strategic
  situations and concluded that in more complex or
  repeated situations or when coordination is
  required the theorys success is more mixed.

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DO PLAYERS IN ACTUAL GAMES PLAY NE STRATEGIES?

• People do poorly at complicated rollback
  reasoning or calculations involving
  probabilities, particularly when they need to
  update probabilities using information revealed
  by actions at earlier stages of the game. They
  fail to take into account subtleties like the
  winners curse.

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DOES RATIONALITY BY ITSELF IMPLY NE?

• We assume that each player behave rationally
  while playing the games. This assumption has been
  criticised by many psychologists and behavioural
  scientists.
• The assumption of rationality alone is not enough
  to establish the case for NE.

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DOES RATIONALITY BY ITSELF IMPLY NE?

• Given the strategies and payoff of a game a
  player can calculate its NE. But if he has
  specified the game incorrectly, his calculation
  may lead him to a strategy that is not correct
  for him to play in the true game.

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DOES RATIONALITY BY ITSELF IMPLY NE?

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IS THE EXPECTED PAYOFF MAXIMIZATION ASSUMPTION
REASONABLE?

• The whole framework of Game Theory has been based
  on the assumption that the players objectives
  are their expected payoffs.
• Payoffs do not have to be measured in money
  units a non-linear scale of payoffs can capture
  a players aversion to risk

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IS THE EXPECTED PAYOFF MAXIMIZATION ASSUMPTION
REASONABLE?

• Consider any zero-sum game in which we have two
  poor strategies. Lets call the call the
  relatively safe strategy (the percentage play) P
  , and the more risky strategy (the non-percentage
  play) R. The opponent has two poor strategies,
  also P and R his P is best response to our P and
  his R to our R. The figure on the next slide
  shows the table of our play succeds these are
  not our payoffs.

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IS THE EXPECTED PAYOFF MAXIMIZATION ASSUMPTION
REASONABLE?

• agtbgtcgtd
• The risky play does really well if the opponent
  is not prepared for it (our success probability
  is A), but really badly if he is (our success
  probability is a D), while the percentage plays
  does moderately well in either case (you succeed
  with a probability of b or c), but a little worse
  if the opponent expects it (cltb).
• Let our payoff utility be W if our play succeeds
  and L if it fails.A really big occasion is when
  W is much bigger than L.

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IS THE EXPECTED PAYOFF MAXIMIZATION ASSUMPTION
REASONABLE?

• OPPONENT EXPECTS
• MY PLAY

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IS THE EXPECTED PAYOFF MAXIMIZATION ASSUMPTION
REASONABLE?

• Our expected payoff is bW(1-b)L. This is a
  zero-sum game,so the opponents payoffs in each
  cell are just the negative of ours.
• In the mixed strategy equilibrium our probability
  p of choosing P is defined by
• P(a-d)\(a-db-c)
• The opponents q-mix to find
• q(a-c)\(a-cb-d)

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IS THE EXPECTED PAYOFF MAXIMIZATION ASSUMPTION
REASONABLE?

• OPPONENT EXPECTS
• MY PLAY