{"id":590,"date":"2023-05-22T00:08:30","date_gmt":"2023-05-21T15:08:30","guid":{"rendered":"https:\/\/saraheee.com\/?p=590"},"modified":"2023-05-27T21:44:23","modified_gmt":"2023-05-27T12:44:23","slug":"game-theory-1-expected-utility-theorem","status":"publish","type":"post","link":"https:\/\/saraheee.com\/ko\/2023\/05\/game-theory-1-expected-utility-theorem\/","title":{"rendered":"Game Theory 1 – chap01. Expected utility theorem"},"content":{"rendered":"

LECTURE 1: NORMAL-FORM GAMES (1)<\/strong><\/h2>\n\n\n\n

What is Game Theory?<\/h3>\n\n\n\n

Game Theory models situations in which multiple players make strategically interdependent<\/mark><\/strong> decisions.
e.g. 1, goods \\(x_{1}, x_{2}\\) prices \\(p_{1}, p_{2}\\)
how much the user will consume \\(x_{1}\\) and \\(x_{2}\\) respectively u(\\(x_{1}, x_{2}\\))
\u2192 This ‘classic theory’ is ‘independent’.<\/mark><\/p>\n\n\n\n

e.g. 2, interdependent decisions: \\(u_{1}(a_{1}, a_{2})\\) and \\(u_{2}(a_{1}, a_{2})\\)
That is, your outcomes depend both on what you do and what others do.<\/p>\n\n\n\n

Examples abounds:
poker, chess, most sports games, negotiations, bargainings, auctions, contracts, contests, partnerships, international relations,
trade agreements, regulations, procurements, electoral campaigns, etc.<\/p>\n\n\n\n

The theory can be classified into
(1) Noncooperative game theory<\/strong><\/mark>
: Focus on individual decision making in strategic settings and make a prediction.
(2) Cooperative game theory<\/mark><\/strong>
Focus on the coalition individuals may form and predict what coalitions will form.
(e.g., player 1, 2 of player 1-3 are united(coalition) – individual decision making(X). collective decision making(0))<\/mark>
+ a wedding market<\/mark><\/p>\n\n\n\n

Taxonomy<\/h3>\n\n\n\n

Games are typically categorized into 4 classes:
information ‘complete\/incomplete’, players ‘simultaneous\/sequential’ decision making.<\/p>\n\n\n\n

    \n
  1. Static games with complete information: complete, simultanious<\/mark><\/strong>
    players act simultaneously with complete information about rules of the game being played. – the most fundamental’s game.<\/li>\n\n\n\n
  2. Dynamic games with complete information: complete, sequential<\/mark><\/strong>
    players act in a givin order with complete information about the rules, but with or without observation of other player’s behavior (e.g., chess or go)<\/li>\n\n\n\n
  3. Static games with incomplete information: incomplete, simultanious<\/mark><\/strong>
    (e.g., bargaining or auctions)<\/li>\n\n\n\n
  4. Dynamic games with incomplete information: incomplete, sequential<\/mark><\/strong>
    (e.g., regulations, procurements, contracts)<\/li>\n<\/ol>\n\n\n\n

    Games are a branch of microeconomics, where “microeconomics” refers to individual choice and games are inherently uncertain.<\/mark><\/p>\n\n\n\n

    Games as a Choice Problem under Uncertainly<\/h3>\n\n\n\n

    First and foremost, every game features strategic interdependence: \\(u_{1}(a_{1}, a_{2})\\) and \\(u_{2}(a_{1}, a_{2})\\)
    When \\(a_{2}\\) is not observed, player 1’s optimal behavior would depend on what s\/he believes player 2 will do
    For the reason, most games theory books begin with choice theory under uncertainty.<\/p>\n\n\n\n

    Preference and Utility<\/h3>\n\n\n\n

    e.g., graph \\(p_{1}x_{1} + p_{2}x_{2} \\leq I\\), x in budgets<\/p>\n\n\n\n

    “x \u227f y” means that DM thinks x is at least as good as y. (DM: decision maker)
    Classic choice theory stems from the next 3 assumptions on \u227f
    (1) \u227f is complete (2) \u227f is transitive (3) \u227f is continuous: if for all \\(x_{n}\\) \u2192 x and \\(y_{n}\\) \u2192 y with \\(x_{n} \u227f y_{n}, x \u227f y\\)<\/p>\n\n\n\n

    description of (1): for all x and y in X, DM can tell x \u227f y, y \u227f x or x\u227f y and y \u227f x
    (\u2190 x and y are indifferent. \u21d4 x \u223c y)
    description of (2): x, y, z in X, if x \u227f y and Y \u227f z, the result is x \u227f z
    description of (3): complete(1) is required for function. transitive(2) is required because if u(x) \u2265 u(y), u(y) \u2265 u(z)), then u(x) \u2265 u(z) in a real number.
    If \u227f satisfies the three axioms above, there exists a continuous utility function u(\u30fb) which represents DM’s preferences \u227f, that is, x \u227f y if and only if u(x) \u2265 u(y).
    (Expressing that the utility function represents the DM’s preference exactly)<\/mark><\/p>\n\n\n\n

    Choice under Uncertainty<\/h3>\n\n\n\n

    EXAMPLE 1.1 (Coin-Toss Gamble)<\/mark>: You can choose either participate or not. If participate, a fair coin will be tossed ~.
    Action N results in a certain outcome = no change in wealth
    Action P results in uncertain outcomes = a change in wealth<\/p>\n\n\n\n

    [Basic Elements] – To describe a decision problem under risks,
    1) Outcome space = the set of possible outcomes \/ In the coin-toss gamble, (0, +10, -10)
    2) Lottery = a probability distribution over outcomes \u21d4 an action \/ Action P \u21d4 (0, 1\/2, 1\/2)
    \u2192 Action N \u21d4 (1, 0, 0)
    3) Prospect = possible outcomes + prob dist’n \/ ActionP \u21d4 (+10, -10; 1\/2, 1\/2) – possible outcome, excluding 0<\/mark>
    \u2192 Action N \u21d4 (0, 1)
    \u227f over uncertain outcomes \u21d4 \u227f over lotteries or prospects<\/p>\n\n\n\n

    EXAMPLE 1.2<\/mark>: You can choose between two bets, without knowing the face of a dice:
    Bet I yields 50 if the dice shows 1, 2, 3; 100 if 4 or 5; and 200 if 6.
    Bet 2 yields 50 if the dice shows an odd number and 100 if an even number.<\/p>\n\n\n\n

    sol) (50, 100, 200), Bet1 \u2194\ufe0e (1\/2, 1\/3, 1\/6) = \\(L_{1}\\), Bet2 \u2194\ufe0e (1\/2, 1\/2, 0) = \\(L_{2}\\)
    Bet1 \u227f Bet2 \u21d4 (1\/2, 1\/3, 1\/6) \u227f (1\/2, 1\/2, 0)
    assunption) The \u21d4 above is from consequentialism: people care only about final outcomes.<\/p>\n\n\n\n

    e.g., If Bet 3 coin head 50, tail 100 \u2192 (50, 100, 200) \u2192 (1\/2, 1\/2, 0) = \\(L_{2}\\) = \\(L_{3}\\), so it is indifferent to Bet3
    \u2192 Bet2 \u223c Bet3<\/p>\n\n\n\n

    Expected Value<\/h3>\n\n\n\n

    One may argue, Bet1 \u227f Bet2 because
    Bet1: \\(\\frac{1}{2}\\)\u30fb50 + \\(\\frac{1}{3}\\)\u30fb100 + \\(\\frac{1}{6}\\)\u30fb200 = \\(\\frac{275}{3}\\)
    Bet2: \\(\\frac{1}{2}\\)\u30fb50 + \\(\\frac{1}{2}\\)\u30fb100 + 0\u30fb200 = 75<\/p>\n\n\n\n

    EXAMPLE 1.3 (ST. Petersburg Paradox)<\/mark>: A coin is flipped until a head appears.
    If a head first appears on the n th flip, you are paid \\(2^{n}\\).<\/p>\n\n\n\n

    sol) H \u2192 2\u30fb\\(\\frac{1}{2}\\) = 1, TH \u2192 4\u30fb\\(\\frac{1}{4}\\) = 1, TTH \u2192 8\u30fb\\(\\frac{1}{8}\\) = 1, so expected value: 1 + 1 + … \u21d2 \u221e<\/strong>
    \u2192 Just considering the expected value alone<\/strong><\/mark> is not enough to determine what type of bet one likes.<\/p>\n\n\n\n

    Bernoulli’s Solution to the Paradox<\/h3>\n\n\n\n

    Individuals respond not to the dollar prize of a gamble but to the utility from the prize:
    Bet1 \u227f Bet2 if \\(\\frac{1}{2}\\)\u30fbu(50) + \\(\\frac{1}{3}\\)\u30fbu(100) + \\(\\frac{1}{6}\\)\u30fbu(200) \u2265 \\(\\frac{1}{2}\\)\u30fbu(50) + \\(\\frac{1}{2}\\)\u30fbu(100) + 0\u30fbu(200),
    that is, the expected utility from Bet1 \u2265 the expected utility from Bet 2.
    Now the follow-up question is whether such a utility function exists.<\/p>\n\n\n\n

    Expected Utility Theorem<\/h3>\n\n\n\n

    THEOREM 1.1 (Von Neumann and Morgenstern(1947))<\/mark>
    – Book : Theory of Games and Economic Behavior
    – Suppose an individual’s preference over lotteries satisfies completeness, transitivity, continuity, and independence.<\/p>\n\n\n\n

    e.g., 600,000 people die – A can save 400,000 people, B can save everyone with a 1\/3 chance.
    \u2194\ufe0e Out of 600,000 people, if you choose option A’, 200,000 people will die, and if you choose option B’, there is a 1\/3 chance that there will be no casualties.
    Many people choose A and B’, so the assumption of completeness is also debatable<\/mark>.<\/p>\n\n\n\n

    Then there exists a utility function u such that \\(L_{2}\\) \u227f \\(L_{1}\\) \u21d4 \\(\\mathbb{E}[u(L_{2})]\\) \u2265 \\(\\mathbb{E}[u(L_{1})]\\).
    \u2192 If lottery 2 is better than 1, then the expected utility from lottery 2 is greater (the decision maker prefers 2 more (necessary and sufficient condition))<\/mark>
    The function u is called a vNM (or Bernoulli) function. (vNM: von Neumann and Morgenstern)<\/p>\n\n\n\n

    e.g. 1, \\(u(x_{1}, x_{2}) = {x_{1}}^{\\alpha}{x_{2}}^{1-\\alpha}\\) \u2192 \\(log({x_{1}}^{\\alpha}{x_{2}}^{1-\\alpha}) = \\alpha\\log{x_{1}} + (1-\\alpha)\\log{x_{2}}\\)
    \u2192 The two utility funcfions represent exactly the same preferene. (\u2235 log is an increasing function)<\/mark><\/p>\n\n\n\n

    e.g. 2, u of x \u2192 u(x) = x \u2192 Linear. risk-neutral \/ logx : concable \u2192 risk attitude is risk-averse
    \u2192 \u2234 Utility functions on certain keys don’t care if they are logarithmic, but the u function in the expected utility theorem cannot be transformed into logarithms. (\u2235 change in attitude toward risk) \u2192 In this case, we say there is no ordinal property<\/p>\n\n\n\n

    EXAMPLE 1.4<\/mark>. Consider the following two-player game<\/p>\n\n\n\n

    <\/td>L<\/td>M<\/td>R<\/td><\/tr>
    U<\/td>8, 1<\/td>0, 2<\/td>4, 0<\/td><\/tr>
    C<\/td>3, 3<\/td>1, 2<\/td>0, 0<\/td><\/tr>
    D<\/td>5, 0<\/td>2, 3<\/td>8, 1<\/td><\/tr><\/tbody><\/table><\/figure>\n\n\n\n

    Suppose player 1 forms a belief \\(\\theta_{1}\\) = (1\/2, 1\/2, 0) on player 2’s behavior.
    Then player 1 prefers to choose U over D, because<\/p>\n\n\n\n

    sol) \\(u_{1}(U, \\theta_{1}) = u_{1}(U, L)\u30fb\\frac{1}{2} + u_{1}(U_{1}, M)\u30fb\\frac{1}{2} + u_{1}(U, R)\u30fb0 = 8\u30fb\\frac{1}{2} + 0\u30fb\\frac{1}{2} + 4\u30fb0 = 4\\) : expected utility (or) payoff
    whereas \\(u_{1}(D, \\theta_{1}) = u_{1}(D, L)\u30fb\\frac{1}{2} + u_{1}(D, M)\u30fb\\frac{1}{2} + u_{1}(D, R)\u30fb0 = 5\u30fb\\frac{1}{2} + 2\u30fb\\frac{1}{2} + 8\u30fb0 = \\frac{7}{2} = 3.5\\)
    \\(u_{1}(U, \\theta_{1})\\) is called the expected payoff to player 1 from choice U given her belief \\(\\theta_{1}\\).<\/p>\n\n\n\n

    Bayesian Rationality<\/h3>\n\n\n\n

    A decision – maker is said to be Bayesian rational if
    (1) she forms beliefs describing the probabilities of all payoff-relevant events;
    (2) when making decisions, she acts to maximize her expected payoff given beliefs;
    (3) after receiving new information, she updates her beliefs by Bayes’ rule<\/mark> whenever possible.<\/p>\n\n\n\n

    Summary<\/h4>\n\n\n\n

    A review of the microeconomic theory of how to make decisions under uncertainty called game theory.<\/p>\n\n\n\n

    <\/p>\n\n\n\n