From db11dcaf5bbd9dba09ac3310e3cb112eeb3f5e49 Mon Sep 17 00:00:00 2001 From: Martin Stancsics Date: Sun, 6 Oct 2024 18:04:57 +0200 Subject: [PATCH] improve crossref spacing --- src/figures/equilibrium_presentation.py | 2 +- src/presentation/defense.qmd | 78 ++++++++++++------------- src/presentation/include/custom.scss | 10 +++- 3 files changed, 48 insertions(+), 42 deletions(-) diff --git a/src/figures/equilibrium_presentation.py b/src/figures/equilibrium_presentation.py index 4eded02..8f0c360 100644 --- a/src/figures/equilibrium_presentation.py +++ b/src/figures/equilibrium_presentation.py @@ -17,7 +17,7 @@ "benchmark": "pi_P_bench", "bargaining": "pi_P", "pure_retail": "pi_P_noF", - "y_label": "$pi_P$", + "y_label": "$\\pi_P$", }, "fringe-number": { "benchmark": "N_F_bench", diff --git a/src/presentation/defense.qmd b/src/presentation/defense.qmd index 0324435..7d47411 100644 --- a/src/presentation/defense.qmd +++ b/src/presentation/defense.qmd @@ -43,7 +43,7 @@ format: ::: ::: {.fragment} -- The **one indispensable / many smaller plyers** is an economically important special case +- The **one indispensable / many smaller players** is an economically important special case ::: @@ -124,7 +124,7 @@ $\{P, A_1, A_2\}$ | 6 ::: {.notes} - Two ingredients: players (left) and characteristic function (right) - - We don't care about strategies, only about value achieved by coalitions + - We don't care about strategies, only about the value achieved by coalitions ::: @@ -201,13 +201,13 @@ Same idea as with the Nash bargaining solution - Axiomatic reasoning - The Shapley-value is the only sharing rule that satisfies a **nice set of axioms**: efficiency, symmetry, linearity, dummy player axiom [@shapley1953additive] - **Bargaining foundations** - - The value arises as the subgame-perfect equilibrium of various alternating-offer bargaining games [e.g. @gul1989bargaining; @winter1994demand; @hart1996bargaining; @stole1996intra; @inderst2003bargaining] + - The value arises as the subgame-perfect equilibrium of various alternating-offer bargaining games [e.g., @gul1989bargaining; @winter1994demand; @hart1996bargaining; @stole1996intra; @inderst2003bargaining] - A **tractable and sensible-looking** gain-sharing rule - Comparative statics show that it behaves as one would expect ::: {.notes} - You don't need to think about cooperative game theory in the following part - - Think of it as a convenient gain sharing rule with some (cooperative and non-cooperative) micro-foundations + - Think of it as a convenient gain-sharing rule with some (cooperative and non-cooperative) micro-foundations ::: @@ -238,7 +238,7 @@ The value of being indispensable ::: -## Relted literature +## Related literature ::: {.fragment} - **Oceanic games** (theoretical) @@ -282,7 +282,7 @@ $$ - Describes situations, such as - platforms - vertical relationships - - generally any setting with an important large players and many identical small ones + - generally, any setting with an important large player and many identical small ones ::: @@ -783,7 +783,7 @@ Hybrid platforms and bargaining power - Hybrid platforms are - Getting more and more common - Seemingly obvious concerns and high-profile competition policy cases - - Despite this, relatively little reserach + - Despite this, relatively little research ::: ::: {.fragment} - Bargaining between participants is not well understood in the platform setting @@ -791,7 +791,7 @@ Hybrid platforms and bargaining power ::: ::: {.highlight-block .fragment} -Examine the bargaining power implications of hybrid platforms and its welfare consequences +Examine the bargaining power implications of hybrid platforms and their welfare consequences ::: @@ -800,8 +800,8 @@ Examine the bargaining power implications of hybrid platforms and its welfare co ::: {.fragment} - **Shapley-value** based bargaining rule - Analytically more **tractable** - - Has **bargaining microfoundations** [e.g. @hart1996bargaining; @stole1996intra] - - Has precedents in the IO literature [e.g. @hart1990property; @montez2007downstream] + - Has **bargaining microfoundations** [e.g., @hart1996bargaining; @stole1996intra] + - Has precedents in the IO literature [e.g., @hart1990property; @montez2007downstream] ::: ::: {.fragment} - **Abstract away all frictions** @@ -810,7 +810,7 @@ Examine the bargaining power implications of hybrid platforms and its welfare co ::: ::: {.fragment} - Consider a **continuum of small players** case for tractability - - Good approximation of the finite player case even for not too many players + - Good approximation of the finite player case, even for not too many players ::: @@ -938,7 +938,7 @@ The optimal price is an additive markup over marginal costs: ::: -## Free enetry +## Free entry ::: {.fragment} - Potential entrants decide to enter if they can cover the @@ -963,15 +963,15 @@ $$ ## Benchmark model ::: {.fragment .highlight-block .smaller} -**Theorem. 2.1** +**Theorem. 2.1.** The optimal entry fee is given by $$K_F^{opt} = \sqrt{\mu I_F V_F} - I_F$$ ::: ::: {.fragment .highlight-block .smaller} -**Theorem. 2.2** -In the benchmark model under hybrid regime, +**Theorem. 2.2.** +In the benchmark model, under the hybrid regime, - The equilibrium number of fringe firms is decreasing in the platform's product variety: $\frac{\mathrm{d} N_F}{\mathrm{d} N_P} = -\frac{V_P}{V_F} < 0$. - The equilibrium size of the aggregate and consumer surplus are independent of the platform's product variety: $\frac{\mathrm{d} A}{\mathrm{d} N_P} = \frac{\mathrm{d} CS}{\mathrm{d} N_P} = 0$. @@ -1069,7 +1069,7 @@ $$ ::: {.fragment .highlight-block .smaller} **Theorem 2.3.** -In the bargaining model under hybrid regime +In the bargaining model, under the hybrid regime - The equilibrium number of fringe firms is decreasing *fast* in the platform's product variety: $\frac{\mathrm{d} N_F}{\mathrm{d} N_P} < -\frac{V_P}{V_F}$. - The equilibrium size of the aggregate and consumer surplus are decreasing in the platform's product variety: $\frac{\mathrm{d} A}{\mathrm{d} N_P}, \frac{\mathrm{d} CS}{\mathrm{d} N_P} < 0$. @@ -1089,7 +1089,7 @@ Other case: @sec-alternative-param Platform's choice of product variety: @sec-appendix-platform-choice -Extensiions: @sec-application-extensions +Extensions: @sec-application-extensions ::: @@ -1106,7 +1106,7 @@ Other case: @sec-alternative-param Platform's choice of product variety: @sec-appendix-platform-choice -Extensiions: @sec-application-extensions +Extensions: @sec-application-extensions ::: @@ -1114,7 +1114,7 @@ Extensiions: @sec-application-extensions ::: {.fragment} - Tractable model of **hybrid platforms** in which **bargaining** plays a key role - - Applicable to other settings, too (e.g. vertical markets, franchises) + - Applicable to other settings, too (e.g., vertical markets, franchises) ::: ::: {.fragment} @@ -1125,7 +1125,7 @@ Extensiions: @sec-application-extensions ::: {.fragment} - **Policy implications** for such markets - - Apple, Google, etc. having their own apps + - Apple, Google, etc., having their own apps - Microsoft acquiring Activision/Blizzard ::: @@ -1150,14 +1150,14 @@ Characterizing multiplayer free-form bargaining - The **one indispensable player / multiple small players** setting has real-world relevance - Wage bargaining - An inventor with an idea and multiple investors - - A band, where one member owns the PA system + - A band where one member owns the PA system ::: ::: {.notes} - Need some asymmetry for interesting bargaining outcomes - One player being indispensable is - Natural - - Maximally asymmetrix → stronger effect + - Maximally asymmetric → stronger effect ::: @@ -1165,7 +1165,7 @@ Characterizing multiplayer free-form bargaining ::: {.fragment} - Problem: non-cooperative game theory cannot provide predictions without structure - - E.g. timing of the game, who makes the offers + - E.g., timing of the game, who makes the offers - NCGT solution is alternating offer games, but a lot depends on minor details [@hart1996bargaining] ::: @@ -1195,7 +1195,7 @@ Characterizing multiplayer free-form bargaining ::: ::: {.fragment} -- We test, whether: +- We test whether: - Outcomes vary as we would expect, based on bargaining power - Certain CGT solution concepts provide good predictions ::: @@ -1211,29 +1211,29 @@ Characterizing multiplayer free-form bargaining :::: {.smaller} ::: {.fragment} - Early unstructured bargaining papers (1950s-1990s): - - E.g. @Kalischetal1952, @Maschler1965, @NydeggerOwen1974, @RapoportKahan1976, **@MurnighanRoth1977**, @MurnighanRoth1978, @Micheneretal1979, @MichenerPotter1981, **@LeopoldWildburger1992** - - Face to face bargaining, different experimental standards + - e.g., @Kalischetal1952, @Maschler1965, @NydeggerOwen1974, @RapoportKahan1976, **@MurnighanRoth1977**, @MurnighanRoth1978, @Micheneretal1979, @MichenerPotter1981, **@LeopoldWildburger1992** + - Face-to-face bargaining, different experimental standards ::: ::: {.fragment} - Free-form bargaining - - E.g. @Galeottietal2018, @HossainLyonsSiow2020, **@NavarroVeszteg2020** + - e.g., @Galeottietal2018, @HossainLyonsSiow2020, **@NavarroVeszteg2020** - Almost always bilateral ::: ::: {.fragment} - Multi-lateral bargaining - - E.g. @MonteroSeftonZhang2008, @MitsutsuneTakanori2014, @TremewanVanberg2016, @chessa2023experiment, **@ShinodaFunaki2022** + - e.g., @MonteroSeftonZhang2008, @MitsutsuneTakanori2014, @TremewanVanberg2016, @chessa2023experiment, **@ShinodaFunaki2022** - Structured or semi-structured ::: ::: {.fragment} - Fairness views in bargaining - - E.g. @LuhanPoulsenRoos2019, @Schwaninger2022, **@ClippelRozen2022** + - e.g., @LuhanPoulsenRoos2019, @Schwaninger2022, **@ClippelRozen2022** ::: :::: ::: {.notes} - @MurnighanRoth1977: shoes game ($Y=0$); series of papers: information and communication structure are important - @LeopoldWildburger1992: face to face, very long rounds, convex comb. of N + ES or SV + ES -- @NavarroVeszteg2020: bilateral, chat, tests axioms (rejets scale invariance) +- @NavarroVeszteg2020: bilateral, chat, tests axioms (rejects scale invariance) - @ShinodaFunaki2022: chat in some treatments, semi-unstructured; how existence of core affects outcomes - @ClippelRozen2022: outside party assigns payoffs, SV + ES ::: @@ -1267,7 +1267,7 @@ How to divide the value between the players? ::: {.notes} - Both are increasing in Y - - Nucleolus is flat for low Y (no real bargaiing power) + - Nucleolus is flat for low Y (no real bargaining power) - SV predictions are less extreme - Core would contain ES for Y < 67 ::: @@ -1338,7 +1338,7 @@ How to divide the value between the players? ## Testing the axioms {#sec-axioms} ::: {.fragment} -- Axioms charactetrizing the Shapley value and the core +- Axioms characterizing the Shapley value and the core - Look at observed **behavior** - **Survey** at the end of the experiment ::: @@ -1377,7 +1377,7 @@ Details: @sec-axioms-appendix - Although **more** in treatments **with high bargaining power disparity** - Discussion **about the experiment** itself - Players can clarify the rules for others - - Provides feeddback for the experimenters + - Provides feedback for the experimenters ::: :::: @@ -1444,7 +1444,7 @@ $$ ::: ::: {.fragment .smaller} -May be multi-valued, e.g.: +It may be multi-valued, e.g.: | $\pi_P$ | $\pi_{A_1}$ | $\pi_{A_2}$ | |:--------:|:-----------:|:-----------:| @@ -1487,7 +1487,7 @@ $\{P, A_1, A_2\}$ | 6 ::: {.column width="50%"} -Maximize smallest excess +Maximize the smallest excess ::: {.r-stack} @@ -1579,7 +1579,7 @@ $\{P, A_1, A_2\}$ | 6 ::: {.column width="58%"} ::: {.highlight-block .fragment .smaller} -**Theorem 1.2** +**Theorem 1.2.** Let $f$ be continuous on $[0, 1]$. Then @@ -1647,9 +1647,7 @@ Weighted value: @sec-multiple-big-players ## Heterogeneity – general case {visibility="uncounted" #sec-heterogeneity-general} ::: {.highlight-block .smaller} -**Lemma 1.2** - -Assume that +**Lemma 1.2.**. Assume that - $X_n \xrightarrow[]{d} X$ where $X$ with cdf. $H$. - The whole probability mass of $X$ is concentrated on the manifold $(a_1(s), \dots a_L(s)), t \in [0, 1]$. @@ -1718,7 +1716,7 @@ $$ ::: {.smaller} - Higher platform product variety → lower consumer surplus **when platform is operating in hybrid mode** - Does it want to have more products? -- Assume that at time 0 the platform can invest in own products at cost $I_P$ per product +- Assume that at time 0, the platform can invest in its own products at cost $I_P$ per product ::: ::: {.fragment .smaller .highlight-block} diff --git a/src/presentation/include/custom.scss b/src/presentation/include/custom.scss index c282035..9f7627c 100644 --- a/src/presentation/include/custom.scss +++ b/src/presentation/include/custom.scss @@ -27,9 +27,17 @@ a.title-link { } -.reveal .crossref { +.reveal div.crossref { text-align: right; font-size: large; + position: fixed; + bottom: 5px; + right: 0px; +} + +.reveal .crossref p { + margin-top: 0px; + margin-bottom: 0.2em; } .reveal div.slide-number {