Game Theory 1 – chap01. Expected utility theorem

LECTURE 1: NORMAL-FORM GAMES (1)

What is Game Theory?

Game Theory models situations in which multiple players make strategically interdependent 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(\(x_{1}, x_{2}\))
→ This ‘classic theory’ is ‘independent’.

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.

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

The theory can be classified into
(1) Noncooperative game theory
: Focus on individual decision making in strategic settings and make a prediction.
(2) Cooperative game theory
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))
+ a wedding market

Taxonomy

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

  1. Static games with complete information: complete, simultanious
    players act simultaneously with complete information about rules of the game being played. – the most fundamental’s game.
  2. Dynamic games with complete information: complete, sequential
    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)
  3. Static games with incomplete information: incomplete, simultanious
    (e.g., bargaining or auctions)
  4. Dynamic games with incomplete information: incomplete, sequential
    (e.g., regulations, procurements, contracts)

Games are a branch of microeconomics, where “microeconomics” refers to individual choice and games are inherently uncertain.

Games as a Choice Problem under Uncertainly

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.

Preference and Utility

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

“x ≿ 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 ≿
(1) ≿ is complete (2) ≿ is transitive (3) ≿ is continuous: if for all \(x_{n}\) → x and \(y_{n}\) → y with \(x_{n} ≿ y_{n}, x ≿ y\)

description of (1): for all x and y in X, DM can tell x ≿ y, y ≿ x or x≿ y and y ≿ x
(← x and y are indifferent. ⇔ x ∼ y)
description of (2): x, y, z in X, if x ≿ y and Y ≿ z, the result is x ≿ z
description of (3): complete(1) is required for function. transitive(2) is required because if u(x) ≥ u(y), u(y) ≥ u(z)), then u(x) ≥ u(z) in a real number.
If ≿ satisfies the three axioms above, there exists a continuous utility function u(・) which represents DM’s preferences ≿, that is, x ≿ y if and only if u(x) ≥ u(y).
(Expressing that the utility function represents the DM’s preference exactly)

Choice under Uncertainty

EXAMPLE 1.1 (Coin-Toss Gamble): 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

[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 ⇔ an action / Action P ⇔ (0, 1/2, 1/2)
→ Action N ⇔ (1, 0, 0)
3) Prospect = possible outcomes + prob dist’n / ActionP ⇔ (+10, -10; 1/2, 1/2) – possible outcome, excluding 0
→ Action N ⇔ (0, 1)
≿ over uncertain outcomes ⇔ ≿ over lotteries or prospects

EXAMPLE 1.2: 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.

sol) (50, 100, 200), Bet1 ↔︎ (1/2, 1/3, 1/6) = \(L_{1}\), Bet2 ↔︎ (1/2, 1/2, 0) = \(L_{2}\)
Bet1 ≿ Bet2 ⇔ (1/2, 1/3, 1/6) ≿ (1/2, 1/2, 0)
assunption) The ⇔ above is from consequentialism: people care only about final outcomes.

e.g., If Bet 3 coin head 50, tail 100 → (50, 100, 200) → (1/2, 1/2, 0) = \(L_{2}\) = \(L_{3}\), so it is indifferent to Bet3
→ Bet2 ∼ Bet3

Expected Value

One may argue, Bet1 ≿ Bet2 because
Bet1: \(\frac{1}{2}\)・50 + \(\frac{1}{3}\)・100 + \(\frac{1}{6}\)・200 = \(\frac{275}{3}\)
Bet2: \(\frac{1}{2}\)・50 + \(\frac{1}{2}\)・100 + 0・200 = 75

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

sol) H → 2・\(\frac{1}{2}\) = 1, TH → 4・\(\frac{1}{4}\) = 1, TTH → 8・\(\frac{1}{8}\) = 1, so expected value: 1 + 1 + … ⇒
→ Just considering the expected value alone is not enough to determine what type of bet one likes.

Bernoulli’s Solution to the Paradox

Individuals respond not to the dollar prize of a gamble but to the utility from the prize:
Bet1 ≿ Bet2 if \(\frac{1}{2}\)・u(50) + \(\frac{1}{3}\)・u(100) + \(\frac{1}{6}\)・u(200) ≥ \(\frac{1}{2}\)・u(50) + \(\frac{1}{2}\)・u(100) + 0・u(200),
that is, the expected utility from Bet1 ≥ the expected utility from Bet 2.
Now the follow-up question is whether such a utility function exists.

Expected Utility Theorem

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

e.g., 600,000 people die – A can save 400,000 people, B can save everyone with a 1/3 chance.
↔︎ 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
.

Then there exists a utility function u such that \(L_{2}\) ≿ \(L_{1}\) ⇔ \(\mathbb{E}[u(L_{2})]\) ≥ \(\mathbb{E}[u(L_{1})]\).
→ 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))
The function u is called a vNM (or Bernoulli) function. (vNM: von Neumann and Morgenstern)

e.g. 1, \(u(x_{1}, x_{2}) = {x_{1}}^{\alpha}{x_{2}}^{1-\alpha}\) → \(log({x_{1}}^{\alpha}{x_{2}}^{1-\alpha}) = \alpha\log{x_{1}} + (1-\alpha)\log{x_{2}}\)
→ The two utility funcfions represent exactly the same preferene. (∵ log is an increasing function)

e.g. 2, u of x → u(x) = x → Linear. risk-neutral / logx : concable → risk attitude is risk-averse
→ ∴ 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. (∵ change in attitude toward risk) → In this case, we say there is no ordinal property

EXAMPLE 1.4. Consider the following two-player game

LMR
U8, 10, 24, 0
C3, 31, 20, 0
D5, 02, 38, 1

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

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

Bayesian Rationality

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 whenever possible.

Summary

A review of the microeconomic theory of how to make decisions under uncertainty called game theory.

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