Economics and Game Theory in Professional Sports
Class 3: Contest Theory
Simplest Base Setup: Two players, P1 and P2. The probability of P1 winning is P1 = E1 / (E1 + E2), where E = effort of player X. Both P1 and P2 probabilities add up to 1. Unless they put in the same effort, they will win half of the time. The utility function (U) for P1 is U1 = (P1 * V) – CE1, where V = value of winning, CE1 = cost of effort for P1. The effort level that will maximize utility for P1 is C = [E2 / (E1 + E2)2] * V and for P2 is C = [E1 / (E1 + E2)2] * V. Differential ability is if the CE and C for both players are the same, the effort level will also be the same. The effort level for each player is E = V / (4 * C). Two things can be changed in a game to change effort: 1) Changing the value of winning = effort change by > 0, increasing prize value = more effort. 2) Changing the cost of effort.
Szymanski Base Setup
Key differences from the base setup are it is symmetric, there are multiple players (N), and there is discriminating power (gamma, g): how much does effort actually affect the chances of winning? If g = 0, effort does nothing and will not impact chances of winning directly. If g > 0, players put in more effort because effort directly affects winning. The probability that player i wins is Pi = Eig / sum of (Ejg). The utility function of player i is Ei = [V * g * (N – 1) / (C * N)2], saying that the effort level of player i depends on the change in V, g, and CE. The rule is U >= 0, if N, saying that if the number of players is small relative to g, then it is worth participating. If you make it that effort is really effective, you will want to lower the amount of players, N, in that game. Finally, if g, you can have as many players as you want in the game.
Jennifer Brown Base Setup
Takes the perspective that the game is not symmetric (one player is significantly better), where the setup is simplified to N = 2, g = 1 (no discriminating power of effort), and lambda (L) where L > 1 = P1 is better than P2. L = 1 means that both players have equal abilities. The probability of P1 or P2 winning is P1 = L * E1 / (L * E1 + E2), P2 = L * E2 / (L * E1 + E2), where L >= 1. The utility of each player is E2 * V1 = E1 * V2, saying that effort levels are proportional to the value of the prize, and if V is the same for both, E will also be the same. The theoretical structure of this setup is that the more unequal the contest is, the less effort both players will put in.
Class 4: Competitive Balance
Within-season variation – The standard deviation of winning is Owp = SQRT[sum of (WPCTi – 0.5)2 / N], where N = number of teams in the league. If all teams are equal, the expected variation is Oe = 0.5 / SQRT(G), where G = number of games each team plays. The non-Scully measure is NS = Owp / Oe. For reference, NS of the most competitive leagues are soccer and hockey with NS = 1.5~, middle range with football and baseball, and least competitive with basketball with NS = 2.5~. An alternative measure for within-season is what range of the season is a team competitive? For year-to-year competitive balance, measure the correlation of wins or winning percentages from one year to the next. For between-season competitive balance (long-run), use HHI to measure the significance of championships. HHI = sum(ci / T)2, where c = number of championships won for team i, and T = total championships won within that range. Example: 10 teams won 1 each = (1 / 10)2 * 10 = 1 / 10.
Class 5: Team Profit and Valuation
Sources of revenue are ticket sales, broadcasting rights (top 2), concessions/venue revenue, licensing income, and intra-league transfers. Team revenue vs. shared revenue: Team revenue for MLB are ticket sales and local TV rights. The NFL has no individual team revenue. Shared revenue for MLB is national TV rights. For the NFL, shared revenue is NFL ticket sales and national TV rights. Implications of shared revenue/team revenue matters when asked if teams want to win or maximize profitability: As an NFL team, there are not many options/financial levers to maximize the profitability of your team. Compared to MLB, teams that spend more money to increase the percentage of winning also increase the fan base, which creates incentives to increase money generated from team revenue sources. Therefore, winning in the NFL is just for fun; winning does not change much profitability, especially in the short-run, whereas MLB or the Premier League is profitable. Different sources of revenue are important to different leagues in different ways: The NFL is all about national TV; therefore, the goal is to increase the number of fans watching on TV. MLB is all about local markets (local TV rights and ticket sales). From a player valuation POV, the NFL trying to recruit the best player is not about profit anymore, whereas MLB bringing in the best player could bring more fans, which changes the local market (potentially more revenue). One problem of revenue sharing is teams in MLB can still make money for being bad (no ticket sales) and get money from intra-league transfers. Costs of teams and leagues are basically 50% of revenue goes to payroll. The NBA, NFL, and NHL have union contracts that help negotiate that percentage. MLB has no union contract but still devotes 50% of revenue to payroll. We do not know the other 50% of costs; this is why public companies cannot invest/buy teams because it would release financial information. Average profits of a team matter, especially in the NFL since it is seen as a target for players and union negotiations. Unions and players threaten the league, saying that if you don’t pay us more, we will shut down the league, and you won’t make that $100M a year. The NFL union is very weak. The NFL is different from other leagues in 3 ways: 1) The average NFL career is 3 years, 2) How many players actually improve their career/position or become superstars?, 3) NFL team roster size is huge. This is why it is difficult to create a policy/rule for the free-agency structure in the NFL because the 8 backup players would rather get their guaranteed $600K/year and not worry about getting $10-15M in free agency because they know they’ll never get that. Yearly profit as an indefinite annuity: The PV of a constant revenue stream is PV = R / (1 – d), where R = yearly revenue, d = discount rate which is d = 1 / (1 + r) where r = interest rate. Discount rates and interest rates are different since discount rates embed other factors like opportunity cost. Example: A team has $100M revenue yearly with a discount rate of 0.95; the team is then worth = ($100M / (1 – 0.95)) = $2B. If a team is sold for more than its valuation, what are we missing? 1) Maybe the discount rate is either too high or too low, i.e., the Broncos were sold for $4.5B. A team with $100M revenue divided by 0.02 would have a valuation of $5B, but this 2% interest rate is unreasonable for someone with $5B to make that investment, 2) Maybe we do not understand the value and profitability of the Broncos as much as we like, i.e., taxes. Growing profitability: The PV of a team with growing profitability each year is PV = R + d(1 + g) * R + d(1 + g)2 * R + d(1 + g)3 * R +… where g = growth rate. The idea behind growing profitability is the g of a team acts to offset some of the discount, d. If d = 0.95 (i = 5%~) but g > 5% then the indefinite stream of profits would be infinite, as long as g < i we have indefinite streams of profit. Owners buying teams in small pro leagues are betting on the growth rate of that team. This creates the question of how much team valuations are a portion vs. interest rate. > EGO rents: The PV of a revenue stream with ego rent is PV = (R + ego) + d(1 + g)(R + ego) + d(1 + g)^2(R + ego) +… General PV of a team with ego and growth is PV = (R + ego) + d(1 + g)(R + ego). Profit or win maximization? Teams making money vs. profit maximization is NOT the same thing. A team can be trying to make a profit but is failing at it. What owners are trying to do can impact league policies heavily: In terms of a revenue-sharing policy, the consequences of revenue sharing depend on whether owners are profit-maximizing or winning. If trying to max profits, revenue sharing does not affect the behavior of the team since the profit-maximizing thing to do is to pocket that shared revenue, whereas maximizing wins with revenue sharing, teams now have more money to spend on themselves to make them better, which improves competitive balance (improving comp balance with shared revenue only works if teams want to win). The purpose of revenue sharing is not for comp balance rather for people to own teams in small market areas. In terms of lottery draft policy, if teams are maximizing profit, then the draft would only shift the value of the #1 draft pick from a bad team to a good team. If teams are maximizing wins, drafting the best player would generate more revenue for me. Owners that maximize wins have more options with the draft than teams that maximize profits.
Class 6: Socioeconomic Valuation
Contest theory review: A method of thinking, what does it mean for a league to be balanced? Balanced percentage of winning (competitive balance) or balanced ability of players? Think of the link between the percentage of winning and ability levels and the gamma. Two arguments for comp balance: 1) Need comp balance for more fans to watch (more profit), 2) 90% of other things justified by comp balance also have side effects of increasing profit (i.e., salary caps). Sports attendance and viewership are a luxury good (as income rises, so does consumption) and technological innovations matter a lot (affects viewership).
Class 9: Game Theory
Intro to game theory: Static vs. dynamic games, where static is simultaneous games (soccer shootout) and dynamic is sequential games. Complete vs. incomplete information. Performance-enhancing drug games (static and complete info) are prisoner’s dilemma games (where payoffs for both players to take the drug simultaneously are lower than if no one took it). Coordination games (multiple equilibria with complete info) are games that have 2 Nash equilibria; the selection of which equilibrium depends on what the scenario is (i.e., car race example). Sequential games (sub-game perfect Nash equilibrium) work backward from the last decision (backward induction). Example: 2 players, 9 sticks, each player can take 1, 2, or 3 sticks, and the player that picks up the last stick wins. In the Nash equilibrium of this game, is the winner the player that goes 1st or 2nd? Solution: Imagine all 9 sticks laid out and start with the last stick and go back. If there are 1, 2, or 3 left, you win, but if there are 4 left, you lose. If 5 are left, you can win since you would pick up 1, which results in 4 left for the next player. If there are 6 left, you win, 7 left you win, 8 left you lose, and 9, 10, 11 left you win. Sequential decisions in baseball games example: Defense chooses how much to shift; they choose x. The hitter sees x, decides to hit or bunt. Payoff to offense: 1) Hit away: 36 – x or 2) Bunt: 20 + 3x. In the Nash equilibrium, what is the strategy and payoff for both? Solution: If the defense goes first and picks x, the hitter will hit away if 36 – x > 20 + 3x which x < 4. Therefore, the defense picks x = 4 as their first move. The complete strategy for the hitter is to hit if x < 4.
Class 10: Mixed Strategy Game Theory
Zero-sum games (constant-sum) are games where payoffs add up to 100. Zero-sum games have no pure strategy Nash equilibrium and have mixed strategies (the percentage of taking 1 of 2 possible actions). Soccer shootout example above where payoffs = percentage of the goal being scored: What is the pure strategy Nash equilibrium? What is the mixed strategy Nash equilibrium? Solution: For the goalie: Payoff diving left = payoff diving right, 50p + 10(1 – p) = 15p + 40(1 – p) and p = 6 / 13 (percentage shooter kicks left). Now graph both equations and find that the goalie will dive left if p > 6 / 13 or dive right if p < 6 / 13.
Class 11: Monopoly Pricing and Market Segmentation
How much would a profit-maximizing monopolist charge for tickets? Factors that affect the optimal price (other than ticket demand) are capacity constraints, other revenue sources, consumer surplus, effects on winning, and habit formation. The profit function is R = P * Q – MC * Q, where P = price, Q = quantity, and MC = marginal cost. Linear demand example: A team has a demand for tickets of QD = 15000 – 500p. The MC of a fan is $6. What is the optimal ticket price, and how many fans would attend? At the equilibrium, what is the elasticity of demand for tickets? Solution: Inverse demand function to price equation: QD = 15000 – 500p —> 500p = 15000 – QD —> p = 15000 / 500 – QD / 500 simplifies to P = 30 – QD(1 / 500). Graph MC = $6 and the new demand function (P = 30 – QD(1 / 500)). Next, substitute the new demand function and MC of $6 into the profit function to get R = 24Q – Q2(1 / 500). Next, take the derivative of the profit function: 0 = 24 – Q(1 / 250) to get Q = 6000. Now substitute 6000 into the price equation P = 30 – QD(1 / 500) to get P = $18 for 6000 tickets. If the question has capacity, just solve the question as if there was no capacity to get Q to compare. Price elasticity is e = (change in Q / change in P) * (P / Q). e > 1 means elastic demand (e always > 1 since costs are involved) and you would only price tickets in the elastic area of the demand curve. The advanced profit function (with concessions and MC) is R = P * Q – MC * Q + C * Q, where C = average profit from concessions/fan. Empirical evidence that teams price tickets slightly lower than the equilibrium price because it adds consumer surplus (even at max capacity, you can lower the price to increase surplus). Price discrimination: 1st degree (charging people for their WTP), 2nd degree (different prices for the same product, you cannot identify the customer, they identify themselves), 3rd degree (market segmentation, different prices for different markets). For market segmentation, prices vary with the elasticity of demand; more elastic demand = lower prices. Also need market power and the ability to limit resale, and segmentation is heavily impacted by capacity constraints.
Class 12: Pricing Strategies
2nd-degree price discrimination includes quality variation and ticket bundling where we charge different prices for essentially the same product, letting the consumer identify their characteristics and choose which they want. Indirect discrimination examples are volume discounts, souvenir giveaways, and hurdle pricing (coupons). Versioning is indirect price discrimination where versioning is price variation that is > cost variation for different versions of the same product, and pricing depends on incentive compatibility (making sure people buy the option we had in mind for them). Versioning example 1: Two groups of fans (premium and budget) and two types of seats (purple and gold). Premium fans value gold seats at $20 and purple at $10. Budget fans value gold seats at $12 and purple at $8. How much should they charge for gold and purple seats?